Not nearly, actually! Exact tautonyms are pretty uncommon, all things considered. Trinomial tautonyms even less so, like Bison bison bison. I don't think I can give you actual data, but I can say knowing a fair bit of marine taxonomy that it's the exception there.

Edit: It's also prohibited in botany and, I learned today, used to be prohibited for bacteria.

Edit 2: Wikipedia maintains an incomplete list, incidentally.

Edit 3: Forgot to mention I was specifically talking about binomial/trinomial tautonyms. Near-tautonyms are very common as you go up toward the middle of the tree of life, e.g. a family, superfamily, and order sharing basically the same name, just with a different prescribed suffix.