It feels more like an issue on numpydoc's side (regarding init docstring inheritance)?
As I already argued elsewhere I'm not a fan of publically inheriting private classes, but it's not a huge deal either.

It feels more like an issue on numpydoc's side (regarding init docstring inheritance)?

That's half of the story. But even if that was working, do we need a common base class for HPacker and VPacker? If one wants to be completely accurate, one would need slightly different parameter docs for those (width vs. height). These parts of the signature cannot be really well-defined at all in the base class. Making

As I already argued elsewhere I'm not a fan of publically inheriting private classes, but it's not a huge deal either.

Semi-OT: I understand that. It's no problem if the base class does not provide any public functionality (as in this case). In general, I'm wondering how to do this correctly. For example _AxesBase, I don't want to make it public in the sense that users should not directly use it. But I would like the methods of _AxesBase to be available via it's children; in particular also in the documentation.

I don't have a strong opinion in the case of PackerBase. I'm not following up on this.

Note that a common __init__.__doc__ has been set up in #19012; I guess the numpydoc limitation has been addressed at some point in between.