This is a vague question; so vague that I wonder if anyone will get it. Many, perhaps most, equivalence relations that are regularly used in mathematics correspond to the orbits of some group action that is also regularly used. Examples:
- Congruence of integers modulo $m$ corresponds to the orbits of the action $x\mapsto x+m$ of the infinite cyclic group on the integers.
- Similarity of matrices corresponds to the orbits of the general linear group in its action by conjugation on the set of all matrices.
- Consider the symmetric group in its standard action on a set $V$. It also acts on the set of all graphs on $V$ by relabelling of the vertices, and the orbits of this action are the isomorphism classes of graphs.
Of course, given any equivalence relation, we can invent any number of groups whose orbits are the equivalence classes. But in each of the examples the group is not just made up but somehow arises naturally from the problem (told you this is a vague question).
My question is: what equivalence relations are in common use (not just made up now!) that do not correspond to the orbits of a group that arises naturally? I can't define phrases like "arises naturally", but most mathematicians have some intuition concerning what that means so I'm hoping this question is not totally meaningless.