As a discerning dissenting voice, let me say that it is true.
Take abelian $A$. Let $B:=A$ with forgotten enrichment in abelian groups. Then the identity functor is equivalence, but $B$ is not abelian because it is not even additive.
It is all in your definition, doc!! The standard definition of abelian category asks for the category to be pre-additive, which requires enrichment in abelian groups that can be forgotten. Off course, as Fred and Dylan rightly point out, this enrichment is inherent in your category structure.