Sorry for the late answer.

May I point out that the relation "is bimorphic to" is a preorder, that is, a reflexive and transitive binary relation since the identity is a bimorphism and since the composition of two bimorphisms is a bimorphism. 

Every preorder incudes a natural equivalence relation. In this particular context, we may say "$A$ is bimorphically equivalent to $B$" if there exist bimorphisms $A \rightleftarrows B$, for some objects $A$ and $B$. 

One particularly property of such an equivalence relation is that it is weaker than the relation "is isomorphic to" since every isomorphism is necessarily a bimorphism. In category theory, we often encounter situations in which isomorphisms tend to be too strict. Say, considering the equivalence of categories rather than the isomorphism of categories.

You have mentioned balanced categories. In every balanced category, the equivalence relation "is bimorphically equivalent to" becomes precisely "is isomorphic to".

As for the other part of your question, I must say that I am not sure in which geometric context such an equivalence relation would be useful.

Hope this helps!