How to write "represented by" using Mathematical notation? What are possible ways to say "represented by" or "characterised by" in Mathematics? e.g. When a Mathematical object is used to represent or characterise a physical quantity or some mathematical object from a different field:

*

*In Quantum Mechanics: $\doteq$ or $ \equiv$

*In representation theory (matrices representing other mathematical objects).

Is there a universal notation (or notations) proposed which can be used across various fields?
It may be used to relate objects from a different field (physics, etc) to Mathematics, or algebraic notation, or simply a symbolic "notation".
The best I could find so far is $\doteq$ (See below; Sakurai & Napolitano 2nd Ed, page 23, also see ). Others authors have used notations for this purpose for similar reasons.
$$\mid +\rangle\doteq\begin{pmatrix}0\\1\end{pmatrix}$$
I thought maybe there are notations for "represented by" in following areas that can be used outside their context. I think it is useful to know if there are other instances of such usage in following fields:

*

*Isomorphism may imply representations (in representation theorems). In general (e.g for reducible representations) a "represented by" relation may be a homeomorphism or even no morphism (numerical approximation).

*Certain arrow symbols in category theory

*In probability theory, when a random variable is linked to a distribution: $\thicksim$:  e.g. $X_i\thicksim N(\mu_i,\sigma)$

*Approximations in numerical analysis: $\tilde{x}\approx x$, or $\tilde{x_1}=x_1\pm\varepsilon_1$.

*In general, a definition or equality can be seen as a "representation" relationship:
$:=$ or $\overset{\mathrm{def}}{==}$ or $=$.

But above usages do not imply the "represented by" or "characterised by" explicitly. They do so only implicitly, or from the context or the text.
 A: I think that there is a reason that no one has jumped in to answer your question in the way you were perhaps hoping for.
As your examples suggest, "represented" can mean  quite different things in different contexts.
Going along with your examples for the sake of argument, we can all surely agree that

*

*giving a definition of new concept or new notation,

*providing a suitable isomorphism,

*approximating a numerical value

are very different enterprises, and only confusion could possibly arise from assimilating them too closely. So why should we even want a common notation? Isn't part of the point of a good notational system to mark important commonalities and important differences.
The very fact that you struggle to give a generic definition of "representation" which smoothly covers all your cases suggests that, no, we don't have a single notion at work in all these cases. So there's no reason to want a single notation.
A: Part of the problem is that the way you are using "represented by" is not well defined. In fact, in each of the examples you list, different concepts are at play. In some cases, fundamentally different concepts. So it is best to, for any given concept, define the appropriate notion of "represented by" and then introduce appropriate notation. Otherwise, we would end up with two fundamentally different concepts that use the same notation and it would be a nightmare to determine what concept is being referred to. The flexibility mentioned above is not a bug but a feature, though it arguably stems from the ubiquity of meanings of "represent" in ordinary language.
Lets look at some of the examples you list:
(i) Representation Theory: in this context, a "representation" is a technical term and refers to a morphism of algebras $A \to End(V)$ for some algebra $A$ and vector space $V$ (iirc). The reason this concept goes by the name "representation" is that there is an intuitive idea that this morphism lets us think of the elements of $A$ as endomorphisms of $V$ and the multiplication of $A$ as composition. So we can "represent" $A$ as endomorphism of a vector space, use our knowledge of the well understood vector space, and glean properties of $A$.
(ii) Category Theory: in this context, "representation" is most often used in reference to representable functors. This is again a techinical term and refers to a functor $F: C \to Set$ with natural isomorphism $\alpha : F \cong C(c,-)$. What, if any, connection does this have with (i)? Well, in some sense, $\alpha$ allows us to glean properties of $F$ by studying the well (or at least, better) understood hom functor $C(c,-)$. It also goes the other way: $\alpha$ allows us to glean properties of $c$ by studying $F$.
So (i) and (ii) share the following feature: we represent a less understood object in terms of a well understood object so as to glean information about the former. But beyond that, there is not much in common between these two technical uses of "represents".
In (i), we can, in general, only use intuition about $End(V)$ to help conceptually make sense of $A$. But it need not be the case that $A$ has the same algebraic properties as $End(V)$. But in $(ii)$, because of the isomorphism, $F$ and $C(c,-)$ have all the same categorical properties. So we can directly transport results about $C(c,-)$ to results about $F$ using $\alpha$. This is a fundamental difference between the two uses of "represents" and does not have much to do with the colloquial meaning of "represents", but rather has to do with the technical definitions in their respective contexts. So the only thing both (i) and (ii) have in common is that they share some sense of using a well understood object to study another object. And I would say the (i) and (ii) are the mostly closely related of all the examples you list!
Now how does (i) and (ii) fit in with the other examples you list? Not very well, as we can see. And this (along with the differences between (i) and (ii) already mentioned) is ultimately the reason why there is no standardized notation for representation.
Definitional equality: this is not so much a relation of "representing" in the way the earlier ones were. Rather, this is saying the two thing are quite literally identical. So this does not fit into the paradime of "link a poorly understood object with a well understood object", since the objects related by definitional equality are the quite literally the same thing and so we necessarily have the same understanding of both of them.
(iii) Physics examples: when you hear talk about "represent this physical quantity with this mathematical object" I think this should be taken akin to the definitional equality example, with a caveat (though I'm no physicist). For example, we often represent the position of an object with a function. This amounts to saying that the position of the object at $t$ quite literally is the value of the function at $t$. Why does this earn the description "represents"? Because (and this is the caveat) position is not numerical quantity, but the value of the function is a numerical quantity. Rather, the mathematical model, of which the position function is a part, is supposed to describe, i.e. represent, the physical system. And in the mathematical model, position is defined to be the value of (some particular) function. So we say the function represents the position.
So we often think of a mathematical model representing a physical model because the objects in the mathematical model are supposed to correspond to or describe aspects of the physical system. E.g., the derivative of the position function describes velocity. Does this fit in with examples (i) and (ii). Eh, not really. We are not necessarily using a well understood object to understand a poorly understood one. The mathematical model may not be well understood mathematically. So the concept of represents here is fundamentally different than in (i) and (ii). What exactly is the concept of represents being used? * insert philosophy of science here *
(iv) Numerical approximation example: maybe if two values are close enough, we can think of one representing the other. That is, if two values are close enough, then, depending on the context, it may do no harm to work with one instead of the other. Arguably this idea of represents is closer to (i) and (ii) than it is to (iii). Why so? Maybe one of the values is easier to compute so it is nicer to work with. But the similarity ends there. We are not establishing a formal connection between two objects that allows us to rigorously probe a poorly understood object by a well understood one. And, as far as I'm aware, represents is not a techinical term in this context.
So the term "represents" just has too many meanings in natural language for it to have one single technical usage. Thus each discipline will either: (a) define a technical notion of represents (like (i) and (ii) do) or (b) use represents informally (like (iii) and (iv)).
A: Why notation?
The goal of notation is to make it easier to clearly communicate ideas.  You should only use notation when plain language fails to adequately express an idea, when the amount of plain language needed to express an idea becomes cumbersome, or when a particular idea or object is going to be repeated so often that introducing notation will make it easier to follow the exposition.
One should also keep in mind that notation typically increases the cognitive load of reading a document.  Every time you introduce notation, you replace plain language with abstract symbols—these symbols take time to learn and become fluent with.  For example, I think that it is far easier to understand the phrase "a circle in the plane with radius $2$ centered at the origin" than the notation
$$ "\{ (x,y) \in \mathbb{R}^2 : x^2 + y^2 = 4 \}". $$
This doesn't mean that I won't ever use the latter (it is, in some ways, more precise, and there are a lot of contexts in which the reader would have no difficulty understanding the notation), but the former is often preferable.
As notation is used to improve clarity and readability, it is important to consider whether or not introducing new notation will actually help with those goals.
Scope of notation
It is also worth noting that notation is typically locally scoped, i.e. authors will introduce notation which is meant to be understood within the scope of the document they are writing.  Yes, there are examples of notation which are widely understood and standardized (e.g. "$+$" almost always represents an addition operator (though it doesn't always—sometimes it indicates sign, for example) and "$\circ$" typically represents composition (though it can also be understood as the unit "degrees", or to denote the interior of a set, among other uses).
Moreover, authors will often use standard notation in a somewhat non-standard way.  For example, most introductory calculus texts will define a limit as

Let $f$ be a function defined on an open interval $(a,b)$, except possibly at a point $c\in (a,b)$, and let $L\in\mathbb{R}$.  Say "the limit of $f(x)$ as $x$ approaches $c$ is $L$," and write
$$ \lim_{x\to c} f(x) = L $$
if for any $\varepsilon > 0$, there exists some $\delta > 0$ such that
$$ 0 < |x-c| < \delta \implies |f(x) - L| < \varepsilon.$$

This is an entirely reasonable definition for a student who is just learning the material, but it is not the only definition.  There are more general definitions which account for higher dimensional spaces, one-sided limits, and limits in more general topological spaces.  This particular definition is actually quite restrictive, and is generally replaced by a "better" definition in a course on real analysis.
Even "standard" notation sometimes requires some explanation, and any notation needs to be understood in the context in which it is used (e.g. you cannot assume that any notation you use will be understood outside of the document you are writing).
The problem of "is represented by"
As I was writing this, IsAdisplayName posted a very good answer which covers a lot of the concerns that I have about the meaning of "is represented by".  Rather than repeating what they said, I'll give them an upvote and direct you to their answer.
The short version is that the phrase "is represented by" is not a phrase which you have articulated well in your question.  Indeed, you have used this phrase in a number of distinct (and possibly incompatible) different ways.
Because this phrase can mean so many different things in mathematics, it is crucially important to understand the preceding section on scope.  Even if I grant that all of the examples you cite are examples of one object being represented by another (and I have some quibbles about many of them), the examples you give are reasonably scoped within individual documents or specialized branches of mathematics.  Trying to broaden the scope of these notations is likely to cause confusion, and clashes with notation used elsewhere.
For example "$\sim$" in probability theory is used to explain how a random variable is distributed (e.g. $X \sim N(\mu,\sigma)$ means "$X$ is a normally distributed random variable with mean $\mu$ and standard deviation $\sigma$"), while in algebra and topology the symbol "$\sim$" is often used to denote an equivalence relation (e.g. a torus can be understood as $[0,1]^2 / \sim$, where $\sim$ is an equivalence relation which "glues" the sides of the torus together).
Because of this ambiguity of meaning and problem of scope, it seems unreasonable to expect that there will (or should) be a broadly understood, standardized notation for "is represented by".
TL;DR
The phrase "is represented by" is poorly defined and scoped.  It might be very reasonable to introduce notation for this phrase in a particular document, but there should be no expectation that there should be broadly understood and standardized notation for "is represented by".
