Can an algebraic structure have indistinguishable elements? Sometimes, a topological space has indistinguishable points - we call those spaces non-$T_0$. But given such a space, we can always identify indistinguishable points, thereby yielding a $T_0$ space. (Technically, we've taken the Kolomogorov quotient).
Does this sort of thing ever happen in abstract algebra?
Here's two more examples.

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*A preordered set can have comparable, distinct points - in other words it can fail to be antisymmetric. But that's cool, we can identify comparable points to obtain a partially ordered set.

*Sometimes a pseudometric space has distinct points that are zero distance apart. But that's okay, we can identify zero-distance points to obtain a metric space.

Edit: It would be nice to see a definition of 'indistinguishable' for the elements of arbitrary structures. It would then be a consequence of this more general definition that for an arbitrary preordered set $X$ (order relation $\leq$) it holds that $x,y \in X$ are indistinguishable iff $x \leq y$ and $y \leq x$.
Here's an example. Consider the function $f : \mathbb{N} \rightarrow \mathbb{N}$, with $f(n)=0$ for all $n \in \mathbb{N}$. The associated notion of indistinguishability for the structure $(\mathbb{N},f)$ should probably be the relation $\sim$ such that $a \sim b$ iff both $a$ and $b$ equal $0$, or both $a$ and $b$ are distinct from $0$.
Edit2: On the other hand, perhaps it does not make sense to speak of 'the natural notion of indistinguishability in a structure $X$' without first situating that structure in a category. After all, if we're going to quotient out by the indistinguishability relation, epimorphisms will probably show up at some point.
 A: A commutative ring $A$ with $1$ can contain nilpotent elements, which form an ideal of $A$, called the nilradical $nil(A)$ of $A$.  In some contexts, it makes sense to kill off these nilpotents and pass to $A_{red} := A/nil(A)$, the underlying reduced ring.  (Reduced means "all nilpotents are zero".)
A: Yes.  


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*add to a ring some null elements $\eta_i$ with $\eta_i x = 0$ for all $x$.

*add extra coordinates $X_i$ and equations $X_i = 0$ to the presentation of an algebraic variety by equations.

*for an algebraic structure that has a notion of representation, consider elements that act the same in all representations as indistinguishable.  
The quotient implied in the last example is to prevent stupidity like taking a finite ring, adjoining uncountably many null elements, and thinking of that as a "big" structure.
A: $i$ and $-i$ are algebraically indistinguishable points of $\mathbb C$ because conjugation is a field automorphism over $\mathbb R$.
A: Consider the field $\Bbb Q(t,s)$ where $t,s$ are two algebraically independent transcendental numbers.
Then these two numbers are completely inseparable by a first-order formula in the language of fields. 
Generally speaking, if $\cal L$ is some first-order language of some structure, then there are at most $\aleph_0\cdot|\cal L|$ definable elements in any given structure. If by "indistinguishable" we mean "inseparable by a first-order formula with limited parameters$^1$", then any sufficiently large structure will invariably contain a lot of indistinguishable elements.
One good place to learn about these things is model theory, and in particular the concept of "type".
Edit: To your last edit, about $(\Bbb N,f)$ note that $0$ is a definable element of the structure with the formula $x=f(x)$. And since we don't have any other symbols in the language it's really impossible to express anything else. Therefore it's very easy to see that over the empty set, every two non-zero elements satisfy the same formulas with one free variable.
(To see that we can't express anything else, at least without parameters, note that if $m,n$ are non-zero then there is an automorphism which exchanges between the two. Therefore every two non-zero elements must satisfy the same formulas [in one free variable].)

Footnotes:


*

*Of course if we allow any parameter then $\varphi(x,y)$ defined as $\lnot(x=y)$ is sufficient to distinguish between any two members. But if, like in the first example, we allow no parameters - or parameters from a small substructure - then if the universe of the structure is large enough, there will be many indistinguishable elements.

A: A category usually has distinct but isomorphic objects. This generalizes both of your bulleted examples: a preorder is a category in which there is at most one morphism between any two objects, and a pseudometric space is an enriched category (see Lawvere metric space). Getting rid of this extra ambiguity amounts to taking a skeleton. 
Since categories are ubiquitous, this gives a wealth of examples. Here are some which are more algebraic in flavor: 


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*A set $X$ together with an action of a group $G$ may be regarded as a category (in fact a groupoid) with a morphism $x \to y$ for every $g \in G$ such that $gx = y$. Two objects are isomorphic iff they are in the same orbit with respect to the group action. 

*Given a field $k$ we can consider the category of algebraic extensions of $k$. This category contains various algebraic closures of $k$, all of which are isomorphic.

