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.

Hope this helps!