Actually Tartar is correct, at least in this context. Tartar is the medieval latin import word of Tata, which is what the hordes of Genghis Khan called them selves. At the time, it was used as a generic term for all peoples from the eastern steppes. Tatar is a term that refers to a specific turkic ethnic group. I’m not that familiar with this specific time period, but it seems more than possible that the Tartars in the game are not Tatars, if that makes sense.
EDIT: regarding the calling an Irish person English thing It’s actually closer to calling someone from Wales Welsh, instead of Cymry. Tartar is an exonym used by western europeans in the medieval era as a blanket term for all steppe cultures.