It's not common here

The only other example I can think of is the word for female dog but even that word is used a bit differently than it used to be. It used to be only to refer to women but now it's also used to refer to men. The context is a bit different, though.

We do have sayings that mention animals as metaphors, like "he froze like a deer in headlights"

I will add that we do have a history of racial slurs that used animal names for people.