Of course we're animals, but let's use some common sense wrt cultural norms here. A dog of mixed lineage is mutt, but it's completely inappropriate to refer to a multiracial person as such. A female dog is a bitch, a male is a stud; the sexism is pretty obvious when applied to humans. It's fine to talk about owning a dog; it's not ok to talk about owning another human (except perhaps children, in certain contexts).
Yes, we are animals too, but that doesn't mean we should talk about each other in the same way. (And I say this as a vegetarian who thinks we should treat all animals with significantly more respect than we currently do.)
It's completely context dependent; you're right that using male/female is appropriate for humans in certain contexts, e.g., medical usage ("Patient, a 47yo female, presented with..."). But it is
for cultural and historical reasons
generally considered inappropriate to refer to our fellow humans that way in conversation.
Re: mutt, fair enough. Bitch/stud are examples of how animal terms, when applied to humans, take on very different meanings. Purebred is afaik not specific to species, but it is wildly inappropriate to refer to people as such.
At the end of the day, the logic behind what is and is not appropriate has history behind it; animal terms have been used extensively to refer to subjugated peoples; it may be scientifically accurate but that doesn't mean that it's inoffensive.