Yes we have, but it's not a genre.
Action RPGs, Turn-based RPGs, Strategy RPGs, dungeon crawlers, roguelikes, roguelites, etc are all genres.
jRPGs and wRPGs are not, they're regional terms as I said.
Though I might argue wRPG breaks down into North America and not North America (and not Japan) it is still a genre distinction, as much as genre is a useful term anyway. It mostly arose in the 8 and 16 bit eras as far as I can see. Compare something like Might and Magic to Dragon Quest and you have a fairly different style of game. Usually though it refers to the storytelling style -- fairly open and with characters you can bias vs rigid story structure with predefined characters. Or if you prefer one wants to be a computerised version of dungeons and dragons where the other would be a kind of interactive storybook.
It is entirely possible for a dev to make a game in either style regardless of their region of origin -- dragon's dogma being an example of a Japanese dev not using the formula, the eventual fate of the Wizardry series being another interesting one. Equally lines have been very blurred as tech got better and restrictions lifted.
Such things are not limited to games either -- folklore, "magic" (word based, mainly for places with writing and certain religions, vs internal/nature magic), board games (classically when you look at card and board games of East Asia and more recently where you get things like German board games), storytelling styles (see something like Greek tragedy and compare it to things coming out of Japan and possibly also compare those to some of the Norse ones) and more have all had examples of things falling into such regional conventions, which may later be appropriated for other things (Shakespeare borrowed reasonably heavily from a lot of Greek tragedies).