Groups are naturally "the symmetries of an object". To me, the group axioms are just a way of codifying what the symmetries of an object can be so we can study it abstractly.

However, this heuristic breaks down in the case of many abelian groups. Abelian groups more often arise as a "receptacle for addition". What I mean is that they are more intuitively counting combinations of some generating elements. See for example: solution spaces to linear equations, the underlying group of rings and modules, (co)homology groups. They rarely act naturally on anything except themselves, which seems like a copout.

It bugs me that these two intuitions are so far from each other, even though the underlying axioms differ by a single, deceptively-mild assumption. Is there a way to reconcile these two perspectives? Do abelian groups satisfy the group axioms by accident?