He would count as one, I suppose -- being an anthropomorphic character. However (in the way I see it) furries are more or less to do with the fandom. I would say there is a difference depending on your intent. If you are a furry and draw for the sake of producing furry art, then I'd say it's furry. If it's bugs bunny, you could say he's anthropomorphic, but seeing as the illustrator of his character most likely (you never know~) wasn't a furry and didn't intend for the art to be fandom-orientated, Bugs wouldn't be a furry.
I'm not sure if this makes sense to anyone else, but I'm sure it's a mixed opinion and that's fine. This is just mine.
Tl;dr: I think it depends on the intent of the artist on whether it's furry or not.