As Martin Brandenburg and Maxime Ramzi suggest, it is easy to construct examples on small categories.

For example, a one object Ab-enriched category is exactly a ring. The category corresponds to the monoid (multiplication) of the ring and the Ab-enrichment to the addition law. There are monoids which have multiple abelian group structures that make them a ring.

Even more simply, consider the category with two objects $x,y$ and $n$ parallel morphisms $x \to y$. Then all compositions are trivial so an Ab-enrichment is exactly an abelian group structure on $\mathrm{Hom}(x,y)$ of which there are many.

As Martin Brandenburg and Maxime Ramzi suggest, it is easy to construct examples on small categories.

For example, a one object Ab-enriched category is exactly a ring. The category corresponds to the monoid (multiplication) of the ring and the Ab-enrichment to the addition law. There are monoids which have multiple abelian group structures that make them a ring.

Even more simply, consider the category with two objects $x,y$ and $n$ parallel morphisms $x \to y$. We need two morphisms $0, \mathrm{id} \in \mathrm{Hom}(x,x)$ so we may set $\mathrm{End}(x,x) \cong \mathbb{Z}$ as a ring generated by $\mathrm{id}$. Then we set $\mathrm{Hom}(y,x) = 0$ to be the trivial group. The only nontrivial compositions are $\mathrm{Hom}(y, y) \times \mathrm{Hom}(x, y) \to \mathrm{Hom}(x,y)$ and $\mathrm{Hom}(x,y) \times \mathrm{Hom}(x,x) \to \mathrm{Hom}(x,y)$ which must be the unique $\mathrm{Z}$-module structure on any abelian group because the generator $\mathrm{id}$ acts as a unit under composition.

Therefore, an Ab-enrichment is exactly an abelian group structure on $\mathrm{Hom}(x,y)$ of which there are many.

As Martin Brandenburg and Maxime Ranzi suggest, it is easy to construct examples on small categories.

For example, a one object Ab-enriched category is exactly a ring. The category corresponds to the monoid (multiplication) of the ring and the Ab-enrichment to the addition law. There are monoids which have multiple abelian group structures that make them a ring.

Even more simply, consider the category with two objects $x,y$ and $n$ parallel morphisms $x \to y$. Then all compositions are trivial so an Ab-enrichment is exactly an abelian group structure on $\mathrm{Hom}(x,y)$ of which there are many.

As Martin Brandenburg and Maxime Ramzi suggest, it is easy to construct examples on small categories.

For example, a one object Ab-enriched category is exactly a ring. The category corresponds to the monoid (multiplication) of the ring and the Ab-enrichment to the addition law. There are monoids which have multiple abelian group structures that make them a ring.

Even more simply, consider the category with two objects $x,y$ and $n$ parallel morphisms $x \to y$. Then all compositions are trivial so an Ab-enrichment is exactly an abelian group structure on $\mathrm{Hom}(x,y)$ of which there are many.