Identifyng objects in a category Is there a general way of identifying objects in a category to produce a new category? Something like a quotient by a relation on objects. How would the morphisms behave?
It's not hard to compute a few examples of such quotients. Define $C/\sim$ to be the universal category with a functor from $C$ which is well defined on $~$-equivalence classes. This implies a colimit condition on the morphisms. For instance, if $C$ has two distinct objects, each with endomorphisms $\mathbb{Z}$ and with no morphisms between them, then the universal category identifying the two objects of $C$ is the category with one object and endomorphisms $F_2$, the free group on two letters. If instead $C$ had just one morphism between its two objects, and no endomorphisms but the identity, then identifying the two objects would produce $\mathbf{N}$. To get morphisms between distinct objects $[A],[B]$ of $C/\sim$, you'd take all morphisms $A'\sim B'$ with $A\sim A',B\sim B'$ from $C$ with free composition law, modulo the relations coming from $C$. Obviously this is sketchy, but I don't see any impediment to the construction-nor any hope that you'd be able to compute anything of interest in a significant number of cases.
Your question does not make too much sense, categorically speaking. That's because categories are basically collections of morphisms obeying certain composition rules (see for instance the definition given here, or "arrow only categories" in MacLane'CWM). Objects are an "afterthought" and correspond to identity morphisms. So , in this light, your question is: can we identify/merge identity morphisms? But different identities are just morphisms which have neither tip nor tail in common,  so it is a bit "odd" trying to identify/merge them. What about other morphisms which have neither tip nor tail in common, for example?

What is interesting instead, and is done with congruences - as explained by Hayden in his answer - is to identify/merge morphisms which have both tip and tail in common (ie. they belong to the same Hom set). 

Generally speaking, when trying to devise/understand a new construction in category theory, first try to see if it makes sense with morphisms and composition of morphisms, then it will automatically make sense with objects, since objects are nothing but a special kind of morphisms (identity morphisms).