I don’t know enough about these North American lunar names to comment, however, I have noticed them being used more often in the UK in recent years.
This really is a complex subject, however, I think that we should recognise the difference between cultural appreciation, i.e. learning from and acknowledging different cultures and celebrating human beings’ rich heritage, and cultural appropriation which does nothing of the sort, i.e. the very phony, commercialised appropriation of another culture’s ideas, which does nothing to help the communities that birthed them and very often ends up being detrimental or even demeaning to them. As has been said before, Greywolf discusses this in the Bardic course materials.