For some reason I was thinking about this today. There are, in English at least (don’t know about other languages), certain animal-related surnames that are common, and others that just don’t seem to exist at all.
The most obvious: Fox, Wolf, and Bird. Those were the first three that I could think of off the top of my head. Thinking about it some more, I have known people with the last name Badger and Beaver. I’ve also known someone with the last name Deer. I’ve known at least one person with the last name Pigg, though have never seen anyone with just “Pig.” I’ve seen the name Bull, and I’ve seen the name Lyon, but not Lion. (This might be derived from the city of Lyon in France, as some English surnames have French origin, and not have any relation to the animal.)
I have never known, or even heard of, anyone with the last name Dog, Cat, Cow, Horse, Donkey, Squirrel, Mouse, Rat, Bear, Snake (or any sub-variety of snake), or actually any reptile or amphibian at all.
There are, on the other hand, a lot of surnames related to fish. Fish itself is a surname, but also Salmon, Trout, Pike, Bass, and there are probably others I’m not thinking of. I have actually read about people with all of those types of fish as surnames.
Again, I am only talking about the English language here as I am sure there are many animal surnames names in other languages as well, but I’ll leave that to others.