Is there a multivariate alternative to two-sample Kolmogorov–Smirnov test? What I mean is a test that can be used to check whenever two underlying multidimensional distributions differ.
A 2004 article On a new multivariate two-sample test by Baringhaus and Franz maybe helpful, they provided a brief literature review on the two-sample multivariate GoF tests and then a R package
cramer. As the package name suggested their method is related to Cramer's test, a predecessor of Cramer-von Mises.
For one-sample problem Justel et al. developed a generalization of Kolmogorov-Smirnov test. In general it seems the difficulty in multivariate case rooted from extending the definition of EDF (empirical distribution function), so methods based on other measures are worth exploring, e.g. multivariate tests based on ECF (empirical characteristic function) by Fan.
Yes, the two sample KS test was extended to the multivariate case in 2021. https://www.sciencedirect.com/science/article/pii/S016771522100050X
For a practical application of the multivariate KS distance in high dimensions, see my article New Python Library to Evaluate AI-generated Data and Compare Models. I actually created a Python library to do the computations.