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If $C$ is any additive category then every object has a unique structure as an abelian group object so $Ab(C)=C$; but typically $C$ is not abelian. For example, this applies to the category of free abelian groups. One can also think about triangulated categories, which are usually not abelian, although a nice theorem of Freyd gives a canonical embedding in an abelian category. One example that has been studied extensively is the category of spectra in the sense of stable homotopy theory. Similarly, one can consider abelian group objects in the homotopy category of spaces, otherwise known as commutative H-spaces.

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In general, I have already trouble to show that $Hom(A,B)\times Hom(B,C)\to Hom(A,C)$ is linear in the left coordinate