That's the problem. Know-it-alls (and I've done this myself) are quick to point out that apes aren't monkeys. But that distinction is made in the English language, not necessarily in all languages. In taxonomy, there is no monophyletic clade (a clade which includes an ancestor and all of its descendants, and nothing else, of which modern taxonomists are so fond) which includes both old world and new world monkeys, but excludes apes and humans. In other words, taxonomically, there's no such thing as a monkey.
Similarly, people are fond of saying that humans are apes (in more recent years I've heard that humans and apes share a common ape-like ancestor), but if someone says they saw an ape in the zoo, do we assume they might possibly be talking about one of the zookeepers, not in a derogatory manner, but because they're apes too? The word ape is an English one, not scientific nomenclature, and probably didn't originally include humans, and it could be argued still does not in usual discussion, unless one is describing the appearance or manner of a particular human as ape-like. And I've also known people similarly described as hippos and pigs, but doubt that would carry much weight in the halls of natural history either.
The other day, my stepdaughter said 'It's probably not true [there's a fortean in the making with the right guidance], but at school they said people are animals too.' My girlfriend and I agreed they are, like you do. But I mulled this over afterwards. The word 'animal', in general use, is an English word used to describe those creatures which are not humans. Nobody uses it to include people unless they're saying something like, 'humans are animals too', which wouldn't need saying if it was understood that the word included humans.