Assume $(X_i,d_i)$ are polish spaces (or compact metric spaces) for $i=1,2$. Further assume that the 1- Wasserstein spaces $(P_1(X_1),W_1)$ and $(P_1(X_2),W_1)$ are isometrically isomorphic. Does that imply that $X_1$ and $X_2$ are isometrically isomorphic?
The question vaguely connects to the concept of Wasserstein rigidity, i.e., that every isometric automorphism of a Wasserstein space is induced by an isometric automorphism of the base space, which sometimes is true and sometimes not (e.g. the 1-Wasserstein space of the unit interval).
