Attempt to define the notion of subobjects I attempted to generalize the notion of subsets for an arbitrary category.
Given an object $X$, an object $Y$ shall be called $X$'s subobject iff there exists a monomorphism $f : Y → X$.
Dually, $Y$ shall be called $X$'s quotient object iff there exists an epimorphism $g : X → Y$.
These notions don't coincide in general. For example, even in the category $\mathbf{Set}$, $\emptyset$ is a subobject of every sets, but it fails to be a quotient object of every sets.
But I searched on Wikipedia, and apparently there is already the notion of subobjects? To quote:

In detail, let $A$ be an object of some category. Given two monomorphisms:
$u:S→A$ and $v:T→A$
with codomain $A$, we write $u≤v$ if $u$ factors through $v$—that is, if there exists $\phi:S→T$ such that $u=v \circ \phi$. The binary relation $≡$ defined by:
$u≡v \iff u≤v \land v≤u$
is an equivalence relation on the monomorphisms with codomain $A$, and the corresponding equivalence classes of these monomorphism are the subobjects of $A$.

I don't understand the motivation to have this complicated definition. Taken literally, a subobject is a morphism rather than an object.
Is this because my definition is inferior to the standard definition?
 A: Your question has two parts:

*

*Why is a subobject defined as a morphism rather than an object?


*Why is a subobject defined as an equivalence class of monomorphisms rather than an individual monomorphism?
The general concept of subobject in category theory tries to generalize "naive" definitions like that of a subgroup of a group or a linear subspaces of a vector space. Mathematical objects like groups or vector spaces can be regarded as sets endowed with an additonal structure. Okay, that is a bit sloppy, but you certainly get the idea. The formal setting is that of a concrete category:

A concrete category is a pair $(\mathbf C, \phi)$ such that $\mathbf C$ is a category  and $\phi : \mathbf C \to \mathbf{Set}$ (the category of sets and functions) is a faithful functor.
The functor $\phi$ is to be thought of as a forgetful functor, which assigns to every object of $\mathbf C$ its "underlying set" and to every morphism in $\mathbf C$ its "underlying function".

If the functor $\phi$ is amnestic
(i.e. if it has the property that if $f$ is an isomorphism in $\mathbf C$ and $\phi(f)$ is an identity, then $f$ is an identity), we call $(\mathbf C, \phi)$ an amnestic concrete category. Many standard concrete categories are amnestic, for example the categories of groups, of vector spaces and of topological spaces.
Let us adopt the convention that the functor $\phi$ will be omitted from notation. Given an object $X$ of $\mathbf C$, instead of writing $\phi(X)$ we speak about the underlying set of $X$ or simply the set $X$. Each morphism $f : X \to Y$ in $\mathbf C$ can be regarded as a function between the sets $X, Y$; we do not write $\phi(f)$ for this function. A set function from an object $X$ to an object $Y$ is a function $u : X \to Y$ between the sets $X,Y$. We say that a set function $u$  a morphism of $\mathbf C$ if it has the form $u = \phi(\underline u)$ for a (uniquely determined!) morphism $\underline u : X \to Y$ in $\mathbf C$. That is, it can be attributed to each set function $u : X \to Y$ between objects $X,Y$ whether or not it "is" a $\mathbf C$-morphism.
Let us now look at two examples.

*

*Let $\mathbf {Vect}_K$ be the category having as objects all vector spaces over a fixed field $K$ and as morphisms all $K$-linear maps. One defines a (linear) subspace of a vector space $V$ as a non-empty subset $W \subset V$ such that $\alpha_1 w_1 + \alpha_2 w_2  \in W$ for all $w_1, w_2 \in W$ and all $\alpha_1, \alpha_2 \in K$. In that case $W$ can be given the structure of a vector space by using addition and scalar multiplication in the bigger $V$. It is easy to see that this "inherited vector space structure" on $W$ is the unique vector space structure on $W$ making the set inclusion $i : W \hookrightarrow V$ a linear map.


*Let $\mathbf {Top}$ be the category having as objects all topological spaces and as morphisms all continuous maps. One defines a subspace of a topological space $X$ as a subset $Y \subset X$ endowed with the subspace topology inherited from $X$ (i.e. the open subsets of $Y$ are defined as the intersections $U \cap Y$ with open subsets $U$ of $X$). Clearly the set inclusion $i : Y \hookrightarrow X$ is a continous map, but in general there are many toplogies on the subset $Y \subset X$ making $i$ continuous. The distinguishing feature of the subspace topology is that it is the coarsest topology on $Y$ making $i$ continuous.
It turns out that in the above examples $\mathbf C = \mathbf {Vect}_K, \mathbf {Top}$ subobjects $Y$ of an object $X$ can be characterized by the following universal property:
(SO) The set $Y$ is a subset of the set $X$ and for each object $Z$ of $\mathbf C$ and each set function $u : Z \to Y$ the following holds: $u$ is a morphism of $\mathbf C$ if and only if the set function $i \circ u : Z \to X$ is a morphism of $\mathbf C$. Here $i : Y \to X$ denotes the set inclusion function.
I leave the easy proof to you.
Clearly the universal property (SO) can be used in any concrete category $\mathbf C$ to define subobjects. It says that a subobject $Y$ of $X$ is, on the set level, a subset of $X$ and the $\mathbf C$-morphisms $Z \to Y$ are precisely the set functions $u : Z \to Y$ which are $\mathbf C$-morphisms when regard as set functions into the ambient $X \supset Y$.
Taking $Z = Y$ and $u = id$ in (SO), we see that for subobjects $Y$ the set inclusion $i : Y \to X$ is always a morphism of $\mathbf C$. Sometimes this property of $i$ alone suffices to determine the subobject structure of $Y$ uniquely (as for $\mathbf {Vect}_K$), sometimes we need the full power of (SO) (as for $\mathbf {Top}$).
The above definition allows to regard certain subsets of the set $X$ as objects of $\mathbf C$; their "subobject structure" is "inherited from $X$". In general we cannot expect that each subset of $X$ can be endowed with a subobject structure such that (SO) is satisfied. But if such an inherited structure exists, then it is essentially unique. In fact, let $Y_1,Y_2$ denote two objects with the same underlying set which both satisfy (SO). Then the set function $id : Y_1 \to Y_2$ has the property that $i \circ id = i$ is a morphism of $\mathbf C$ which implies that $id : Y_1 \to Y_2$ is a morphism of $\mathbf C$. Similarly $id : Y_2 \to Y_1$ is a morphism of $\mathbf C$. Thus $Y_1, Y_2$ are isomorphic in $\mathbf C$ via the identity function on their underlying set. Note that if $\mathbf C$ is amnestic, then $Y$ is in fact unique.
In non-amnestic concrete categories subobjects are in general not unique. Thus we should either make a choice to assign to adequate subsets of the set $X$ dedicated subobjects (preferably via a specific construction) or regard all isomorphic subobjects as the same object (which means to introduce an equivalence relation for subobjects).
Summarizing, we can introduce in each concrete category a subobject relation $Y \subseteq X$. This means that $Y \subset X$ on the level of sets and that the subobject structure on $Y$ is inherited from $X$. For each subobject $Y$ of $X$ the set inclusion $i :Y \hookrightarrow X$ is a morphisms of $\mathbf C$ and will be called subobject inclusion. Note that a subobject inclusion is not an arbitrary injection which is coincidentally a morphisms of $\mathbf C$, but has to satisfy (SO). Anyway, each object inclusion is a monomorphism in $\mathbf C$ because it is an injection on the level of sets.
This shows that a subobject of $X$ is an object $Y$ of $\mathbf C$ plus a dedicated subobject inclusion $i : Y \to X$ in $\mathbf C$.
In the light of the above considerations the definition of subobjects in a general category is problematic. Of course one is free to define everything, but in my opinion we should try to achieve that a general definition produces the same subobjects in concrete categories as definition (SO). Unfortunately this does not seem to be possible. The approach to use monomorphisms is definitely inappropriate. It works in $\mathbf {Vect}_K$ and many other cases, but for example in $\mathbf {Top}$ monomorphisms are simply continuous injections and the obtained "subobjects" are far from being subspaces in the usual interpretation. Of course one can define subobjects via a certain subclass of all monomorphisms, as done with property (SO), but this requires a preceding idea what the subobjects in the given $\mathbf C$ should be, i.e. again a situational definition.
Let us now explore your "object definition":

Given an object $X$, an object $Y$ shall be called $X$'s subobject iff there exists a monomorphism $f : Y \to X$.

As an example, in the category $\mathbf {Vect}_K$ the monomorphisms are precisely the injective linear maps. Due to your definition each vector space $W$ would be a subobject of each vector space $V$ such that $\dim V \ge \dim W$. But does this give us any useful information? The existence of a monomorphism  is trivial, but we do not know in what way $W$ shall be regarded as sitting inside $V$. There are plenty of subspaces of $V$ having dimension $\dim W$ and all of them are images of $W$ under a linear injection - but your definition does not tell us which subspace shall be specified although that is the essential information.
I think this explains why your "object definition" is not really a good choice. It therefore answers question 1.
Concernig question 2:
There is no real objection to defining a subobject as a dedicated monomorphism. In fact, many authors do so. See for example
Adámek, Jiří, Horst Herrlich, and George E. Strecker. "Abstract and concrete categories. The joy of cats." (2004).
An argument to aggregate monomorphisms to equivalence classes is that there is no practical use to distinguish between equivalent monomomorphisms.
A: You want "An object $B$ and a monomorphism $B\to A$" to be a subobject, but the $B$ is actually entirely superfluous in this definition. Any monomorphism with codomain $A$ must by definition of morphism have a domain as well. So defining a monomorphism as a subobject, rather than an object together with a monomorphism inclusion is more refined, even though it is a smidge more abstract to those who aren't used to it.
As for the equivalence class part, consider as an example the category Set, and in it a set $x$ with one element. How many subobjects does it have? Simple set theory and / or elementary combinatorics say it ought to be $2$. What does your definition say? What is the number of monomorphisms with $x$ as codomain? A lot. However, apart from the single monomorphism $\varnothing\to x$, they all factor through one another and become one big equivalence class. Which takes us back to two subobjects. Which is what we wanted.
A: Monomorphisms indeed quite well model embeddings of subsets.
However, the subset an injective function $A\to B$ designates is actually its range in $B$, and the motivation is that we want to consider two monomorphisms $f:A\to B,\ g:A'\to B$ to determine the same subset of $B$ whenever their ranges coincide.
This can be achieved by defining a preorder on monomorphisms with a fixed codomain:
$f$ factoring through $g$, i.e. $f=g\circ u$ for some morphism $u$, indeed nicely encodes the fact that range of $f$ is contained in range of $g$.
