"Falsehood flies, and truth comes limping after it, so that when men come to be undeceived, it is too late; the jest is over, and the tale hath had its effect: [...] like a physician, who hath found out an infallible medicine, after the patient is dead." —Jonathan Swift
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.