Here's a totally different suggestion, which is why I'm writing it separately. As has already been remarked, Ab has the really nice property that it's enriched over itself and monoidal closed. We don't have this property in Grp because Hom(A, B) doesn't have a natural notion of composition if A != B. One can, however, construct a category with the elements of Hom(A, B) as objects and invertible functors fixing A and B as morphisms, and this category is a groupoid. In other words, we should think of Grp as enriched over Gpd - even better, we should think of Grp as living in Gpd, which is enriched over itself and Cartesian closed. So one reason Grp feels different from Ab is that Grp is neither Cartesian nor monoidal closed.
I don't know whether Gpd feels more similar to Ab than Grp does, though.