Here's how I think about it: if the automorphism group of an object is abelian, this means something very strong about the object. It sort of means you can affect its structure in two different ways, in any order, and they won't affect each other. This to me hints that maybe the object consists of separate "pieces" that can only be affected independently.
The first example like this which comes to mind is a direct (commutative ring) product of finite fields of different characteristics. Each field has a cyclic automorphism group, making it very "simple", and the fields can't map into each other, meaning they can only be affected "independently."
So I'd posit that if you want to think of abelian groups as symmetry groups, you can imagine them as the symmetry groups of objects that "break apart" into "easy" pieces which don't interact with each other (intentionally vague, since I don't want to commit to a particular category).