Here is a shortish answer that relates to several of the above replies: Yes! There is such a theory.
Ideals correspond to a particular type of internal category or groupoid in the category of rings, normal subgroups correspond to `dittos' in the category of groups. Quotients by an ideal/normal subgroup are the coequalisers of the source and target maps of those internal categories (in many settings, quotients may not exist, e.g. this is especially important in geometric cases, hence the theory of Lie groupoids etc.) Both of these, internal categories and coequalisers still work perfectly well in $Cat$, so your impression is not quite right.
For a much lengthier gloss on that:
(i) The peculiarity of the categories of rings and groups (and other similar categories) is that congruences can be replaced by such `normal subobjects'. The congruence is the important thing here not the normal subobject, and in very many situations analogues of lattices of ideals work well. Of course, they are lattices of congruences, and so on.
(ii) For 'quotient', there seems a lot of confusion in terminology. From what you say in the question, I presume that you mean 'quotient ring', for instance, rather than 'ring of quotients'. (Several of the comments seem more in tune with the latter situation.) If that is right then the comment that I made above is relevant. quotients are `really' just coequalisers in a category.
(iii) Turning to $Cat$ itself, congruence relations make perfect sense in $Cat$ and correspond to a particular type of double category. In the category, $Cat/\Sigma$ , of categories with a fixed set, $\Sigma$ of objects, congruences are as you state. In other words, it depends on where one is as to which type of congruence and which type of quotient is appropriate.
(iv) Finally, to answer your specific question: yes, one can look at quotients with respect to congruences of categories or groupoids, that crush objects together. This sort of setting was thoroughly explored by Phil Higgins in his book which was reprinted in TAC a few years ago. (See [[http://www.tac.mta.ca/tac/reprints/articles/7/tr7abs.html]]) That looks at the algebra of groupoids as algebraic objects of a fairly classical nature.
(edit: PS. I had not glanced at the linked Wikipedia article. That is way too restrictive in how it defines the notion of a quotient category, IMHO.)
Take any free ultrafilter $U$ on $(0,1)$, and let $m_U$ be the set of functions which are $0$ "almost everywhere", i.e., on a set in the ultrafilter. It seems to me that is a prime ideal, which is in general not maximal. If all ultrafilter sets have, say, the number 1/2 as a limit point, then $m_U$ is properly contained in the ideal of functions vanishing at 1/2.
Why is it an ideal? That looks easy. E.g., if you have two functions vanishing on $A_1$, $A_2$ respectively, then their sum vanishes on $A_1\cap A_2$, which is again in the ultrafilter.
Why is it prime? Let $f_3:=f_1\cdot f_2$. Let $A_i:= \{x:f_i(x)=0\}$.
Then $f_3\in m_U$ iff $A_3 \in U$ iff $A_1\cup A_2\in U$ iff $A_1\in U $ or $A_2\in U$ iff one of $f_1$, $f_2$ is in $m_U$.
Best Answer
Here is a way to construct a non-maximal prime ideal: consider the multiplicative set $S$ of all non-zero polynomials in $C[0,1]$. Use Zorn lemma to get an ideal $P$ that is disjoint from $S$ and is maximal with this property. $P$ is clearly prime (for this you only need $S$ to be multiplicative.) On the other hand $P$ cannot be any one of the maximal ideals, since it does not contain $x-c$ for every $c \in [0,1]$.