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 particular property of such an equivalence relation is that it is weaker than isomorphism 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, bimorphic equivalence is the same as isomorphism.

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. However, there is one context where biomorphic equivalence is useful.

Bimorphisms in **Top** are precisely the bijective continuous maps. This specifies bimorphic equivalence in **Top**. Let us say that a property $P$ of topological spaces is a *bimorphism invariant* if for every pair $X, Y$ of topological spaces, the statement $P(X)$ and $X$ is bimorphically equivalent to $Y$ implies the statement $P(Y)$. 

Every bimorphism invariant is a topological invariant (an isomorphism invariant in **Top**) since, as I have already mentioned, bimorphic equivalence is weaker than isomorphism.

The converse, that is that every topological 
invariant is a bimorphism invariant does not hold since **Top** is not balanced. However, it turns out that many of the classical topological invariants are in fact bimorphism invariants.

It is enough to consider any [continuous invariant][1] (i.e. property of topological spaces "transferred" by surjective continuous maps) such as compactness, path connectedness, and separability. We can generalise this to other categories and look for isomorphism invariants that are bimorphism invariants in that category.

We thus see that bimorphic equivalence is the right "relaxed" notion of isomorphism.

Hope this helps!

  [1]: https://proofwiki.org/wiki/Definition:Continuous_Invariant