TITLE: Can an algebraic structure have indistinguishable elements?
QUESTION [14 upvotes]: 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.

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.

REPLY [10 votes]: 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: 

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.

REPLY [5 votes]: 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".)