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 the composition of two bimorphisms is a bimorphism.
Every preorder induces a natural equivalence relation (namely, by forcing symmetry). In this context, we may say "$A$ is bimorphically equivalent to $B$" if there exist bimorphisms $A \rightleftarrows B$, where $A$ and $B$ are objects.
One particular property of such an equivalence relation is that it is weaker than isomorphism. In category theory, we often encounter situations in which isomorphisms tend to be too strict, say, when considering the equivalence of categories rather than the isomorphism of categories.
In a balanced category, bimorphic equivalence is the same as objects being isomorphic.
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 that may solidify biomorphic equivalence as being 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 (Top is not balanced). However, it turns out that many of the classical topological invariants are in fact also bimorphism invariants.
It is sufficient to consider any continuous invariant (i.e. a property of topological spaces "transferred" by surjective continuous maps) such as
compactness
path connectedness
separability
etc.
We can generalize this to other categories and look for isomorphism invariants that are bimorphism invariants in that category. For the sake of example, in Ord, the following isomorphism invariants are also bimorphism invariants:
"has a least/greatest element”
“is a totally ordered set”.
Thus, bimorphic equivalence is a good relaxed notion of isomorphisms.
I hope this helps!