I think that'll be his name, just.. the river spirit.

He's not a living deer. He's a spirit. He never strays too far from the pond / stream / idol.

He also doesn't like it when he's moo'd at, especially by fawns. He moo's back at them. His breath is ice cold and very unpleasant to the touch. It's so cold, you can actually see it as it comes from under his mask.