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.