I don't think Yugiri is part of a "beast race" but her people are probably not found on this side of Eorzea, so they're not considered part of the "five races." I assume that when they become playable the "five races" will become six.

As for their name, my guess is Auri. The wind-up Odin minion heavily implies it - the description is an explanation for why Odin, a decidedly Western figure, has a Far Eastern blade, Zantetsuken. It was said that he defeated a Far Eastern "Auri" in battle and took his blade. This Auri was probably a samurai, and we know that Doma is culturally very similar to Japan.

Putting two and two together, that would make Yugiri an Auri.