Maybe this needs a bit more detail, concerning in how far matplotlib.patches.Shadow would not be sufficient.

I fully agree it might just be my lack of understanding of how to use Shadow and would appreciate an example of how a Shadow of the same size as an AnnotationBox may be constructed.

Untested, but AnnotationBox should have a get_bbox_patch() method, that then you can pass to Shadow. A PR with a demo would be most welcome...

There is also matplotlib.patheffects.SimplePatchShadow.
... which makes me wonder if both are really needed.

In any case none of them allow for gaussian bluring - which is in general something for which matplotlib does not have any API.

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3.8 will add configuration of the pie (#25389) and legend (#24666) shadow. I guess this comes down to adding a shadow keyword argument to FancyBboxPatch in a similar manner? In that way one can pass shadow={'ox': 3, ...} as part of the bbox arguments (and this will also be applicable to quite a lot of other things relying on FancyBboxPatch).

This will not handle Gaussian blurring though(?). But that can be added to Shadow separately.