This actually works with one of my current confusions about the new setting set-up: Fey are in charge of the Court of Seasons, but the Primal Spirits are in charge of the natural world.
So, erm, the barbarian player in my setting likes nature, but knows nothing about the Eladrin court of seasons, for some reason.
I think the issue arises from two mythologies that have crossed paths here: Native (North) American-styled spirits of animals (raven, thunderbird, etc.), crossed with Celtic Fey myths. For us there is a cultural divide, but for them the Fey and the Spirits would have served the same purpose: the supernatural.
In a way, this includes Angels and Demons/Devils, and East Asian Oni/Demons.
The supernatural used to be the supernatural, but D&D's taken to the idea of using absolutely everything and fitten them into convenient niches where they can fit. In the original material, this was all there was, really.
Heck, even HP Lovecraft's Mythos was the only Supernatural in his stories, but D&D decided to say "why no, these are Abberations" and pretend that it's perfectly normal and a part of the cosmology we pigeonhole into "the far realm". for the original stories, as you likely know, HPL would have just said that all things supernatural were bizarre and maddening and horrific. Having a class of creatures that were otherworldly but normal or even friendly, would have taken away from the horrific moment of, I guess, revelation of the "truth" he was eventually trying to paint for his readers.
If you're mixing them, you don't really need to change anything per se.
I myself am working on a setting where there is the Mortal Coil and the Supernatural Otherworld. Basically the Fey Market the PCs are in has a lot of creatures who wouldn't be in the normal world; there's a Salamander who sells ritual books, for example, and goblins, and ogres, and all sorts of non-fey who fit there. I don't worry about explaining it, tho some elements (demons and immortals) are going to be introduced later. I just say they're all the folks who are in the Fey Market.