Thanks to character profiles, we know a bit more about a few of the non-Izuna characters: Ashley sells flowers on the street (and is fighting for equality in the world or something). Kiriku is a self-taught swordsman who enters the fighting tournament to help his injured sister. Stein is a robot who wants freedom and searches for his creator. (Izuna, of course, is in the tournament for money.)
It's a good thing story doesn't matter in fighting games, because Windy x Windam is turning out to be a pastiche of fighting game cliches so far. Oh, maybe it's parody!