For axes_grid I think we should consider how to make that a core functionality.

The last example seems to exist solely to derive from that private API, so I don't know what to do about it.

Perhaps that base class should be made public, if we want to support this kind of addition of BoxStyles.

As written elsewhere, I have doubts that private base classes are a reasonable approach. Their code leaks into the child classes, which is intended, but OTOH confusing because you're not seeing the base class. This also messes up sphinx because the inherited methods are not documented and cannot be linked.

Generally, I cannot remember to have seen private base classes anywhere else. I feel it's better to make the base class public, but some of it's members private or abstract.

Note, as part of #16615, the last example was merged into https://github.com/matplotlib/matplotlib/blob/master/examples/userdemo/custom_boxstyle01.py though the private API is still used.

I have an idea of how to get rid of that private base class (essentially, move the mutation aspect handling out of __call__ and into the (only) call site, but as usual this will require ginormous amounts of deprecation machinery. I'll work on it at some point...