Monoids represent maps from some mathematical object to itself. Groups represent the automorphisms of some mathematical object. What do abelian groups represent?
One unsatisfying answer would be that they just represent automorphisms of nice enough mathematical objects. However it seems rather uncommon for the automorphisms of some object to form an abelian group unless they have very few automorphisms. Furthermore, the theory of abelian groups is very different from the theory of groups in general, and abelian groups appear frequently in other applications where they are not the automorphisms of some object in an obvious way, like homology groups.
This suggests that maybe its just a coincidence that abelian groups are groups, and that they represent something else. What do they represent?
I don’t think this is a well formed question. Groups can “represent” all sorts of things. To say that groups in general are just the automorphisms of an object detracts from the abstraction of group theory. Groups can represent automorphisms, but they can represent all sorts of other things. It is often useful to simultaneously consider one group in multiple distinct ways to learn more about them. To reduce that to only one way, automorphisms, is unhelpful.
Now, abelian groups are the same. It’s unhelpful to consider them as one thing. They don’t represent anything in particular, but are instead a useful abstraction of many things at once.
Broadly speaking, this is sort of “the point” of abstract algebra (Topics in Algebra, Herstein). We abstract these algebraic objects away from their specific contexts, and learn much more about them and their connections with other things in the process.