How to Monitor ML Classification Models in production? I've often heard of measures like Population Stability Index and Characteristic Stability Index. I might be mistaken, but these seem to be more applicable towards looking at the changes in univariate distributions and are more linear.

Are there any methods that are more robust in capturing multivariate relationships and interactions?
Yes, PSI and CSI are useful when we want to understand if there is any change/shift in the population type.
But to understand multi collinearity, VIF is what you should look into.
Ref: https://etav.github.io/python/vif_factor_python.html
and if you think you are left with only few variables, then you need to create few additional variables (interaction variables) and where there is a chance that they may not be correlated with others.