As I don't use 'Circle of the' as any actual in-setting organization, the Moon name has never bothered me personally. The same way my games have never had actual bardic colleges that the bards came out of with their abilities, nor especially colleges whose names were in-setting 'The College of Valor' or 'The College of Lore or 'Swords' etc. I've treated them all merely as cool-sounding and interesting out-of-game subclass identifiers, but which have no in-setting place.
I would imagine that if a setting you played in did indeed have "Circles" for the druids to belong to, and those circles in-game were actually called their subclass names... swapping out 'Circle of the Moon' for a different organizational identifier would make all the sense in the world. And if that's the case, then sure, changing the organization's name to the 'Circle of the Primordials' or 'Circle of the Ancients', or 'Circle of the Soul' would work fine.