We should implement a weighted version of the Minkowski distance directly in the MinkowskiDistance class itself consistently with scipy's formulation:

https://docs.scipy.org/doc/scipy/reference/generated/scipy.spatial.distance.minkowski.html

which is different from the old:

https://docs.scipy.org/doc/scipy/reference/generated/scipy.spatial.distance.wminkowski.html

which is already implemented inWMinkowskiDistance.

Note that #21741 targeted for 1.0.2 does a minimal fix for DistanceMetric.get_metric("minkowski", p, w) to do the right think automatically as stopgap compatibility fix to get scipy 1.8 support for scikit-learn 1.0.2 but the long term, maitenance fix is to refactor this to deprecate WMinkowskiDistance completely in scikit-learn 1.1.

See the discussion in #21741 (comment) for more details.