Applications of OT to Algebraic Topology was the subject of my thesis, available here https://github.com/jhmartel/Thesis2019
There remains many interesting questions to solve!
I found the topology of either source or target space was encoded in the topology of the singularity $Z$ of certain $c$-optimal transports from source $(X, \sigma)$ to target $(Y, \tau)$.
In case $Y=\partial X$ , we found the singularity $Z$ had the homotopy type of $Y$ if the cost was attractive (like quadratic cost $c=d^2/2$) and had the homotopy type of $X$ when the cost $c$ was repulsive (see above thesis). We described general technique for building strong deformation retracts in either case.
Topology of $X,Y$ apparently controls the topology of singularities, i.e. nontrivial topology tends to force discontinuous optimal transports.
So I don't think Wasserstein distance captures topology -- rather it is the regularity of optimal transports which captures topology.
For example, one can prove the following: If $X=D^2$ unit disk, with boundary $Y=\partial X=S^1$, and if $X,Y$ are equipped with uniform probability measures, respectively, and if $c: X\times Y\to \mathbb{R}$ is a repulsive cost (satisfying standard (Twist) property and nondegeneracy), then the singularity $Z$ of $c$-optimal transports has the homotopy type of $X$. Therefore $Z$ is nonempty and has exactly one connected component. E.g., $Z$ cannot consist of two discrete points.
This is a weak type of Euler characteristic argument... and it has many similarities to Morse-Floer type homology (in my opinion). There's much more to say, and I have several projects in progress regarding further applications of OT to Algebraic Topology.