When "is" a congruence an object?

Three different situations:

1. Take a group $G$. If you have a normal subgroup $H$, you can form a quotient $G/H$.
2. (Can be considered as a corollary of 1.) Take an $R$-module $M$. If you have a submodule $N$, you can form a quotient $M/N$.
3. Take a topological space $X$. If you have a subspace $U$, you can form a quotient space $X/U$.

However, if you take a (commutative and unital) ring, you need to quotient by an ideal, that (except for the trivial case in which it contains 1) is not a subring.

In general the construction of a quotient object in a given category requires the notion of a congruence, that is very different from a subobject. My question is

How typical is that we can go from a subobject (maybe requiring an additional property) to a congruence?

or more precisely

Is there a categorical notion encompassing this behaviour?

Edit:

(Based on comments by Rob Arthan and GEdgar). First two examples are effective congruences (Definition 1.5 in the link above). In other words we take two identical maps $f\colon A\to B$ and find a pullback $K$ with two maps $K\to A$. It can be treated as a subobject of $A\times A$.

But we want $K$ to be a subobject of $A$, so we could "replace $a\sim b$ by $ab^{-1}\in K$".

from Hot Weekly Questions - Mathematics Stack Exchange