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The answer to both, in biology and evolutionary, YES, we have both.But only in those limited contexts is it true, and even then there are caveats. For instance biology textbooks will tell you that great apes are tailless primates. And in virtually any practical sense that's true. You can't yank a gorilla or a human by the tail. You can't get your tail caught in door. Your pants don't have a hole for your tail to poke through. If you told a child that gorillas have tails, you would be seriously misleading that child even if in some technical sense it was true. You would have to explain that gorillas only have tails in a very specific technical sort of way.
If you aren't trying to be cheeky and entertain students with counterintuitive claims, then you would say that great apes have coccyxes, e.g. tailbones, which are the remnants of lost tails.
How about this: Are whales fish? In the traditional sense, a fish is an animal that lives in the water, so whales (and starfish) are plainly fish. But anybody who has ever heard from marine biologists will exclaim that whales certainly are not fish, they're marine mammals. People will be very annoyed with you if you ever call a whale a fish, they'll take it as an insult to whales. But all mammals, including whales, are tetrapods and descended from an ancient lobe-finned fish. So in a cladistic sense you can claim that whales are fish. In the same sense, humans are also fish. And why not? As you say, we have gills too!