Bio professor here; I've found the danger of anthropomorphism is in interpreting animal body language as if it were human body language. Example: the "smile" on a Rhesus macaque is indeed a sign of an emotion, but it signifies fear/submissiveness, not happiness/friendliness as it would be in a human.
I used to teach elephant behavior and always brought this point up for elephants regarding ears-out, which students tend to misinterpret as friendliness (because it looks like ears-out in friendly dogs) but in elephants, ears-out often means aggression and is often a sign of an impending charge. When we score behaviors, I always had to warn the students "Name the behavior by what it looks like, not by what you think it means" (i.e. call it "ears-out" instead of "friendly") because very often once you get to know that species you realize you've been misinterpreting certain body language cues. Or there are cues that you don't even recognize at first becuase they're so foreign (like: Canada geese do a head-flick move that means absolutely nada to a human - most students don't even perceive that it happened! - but to other geese it's a highly significant "C'mon, let's fly! Let's go!" Over time you start to see these things.). Elephants have a tail-up move that a lot of people miss, and a trunk-to-mouth and trunk-to-temple that can also be really quick.
Historic perspective: in the last 30 yrs, biology has really swung around on the issue of animal cognition and animal emotion. It's now generally regarded as reasonable to assume animals can experience emotions if they have the requisite neural circuitry and they show the corresponding behaviors and choices.
The chief mistakes that I find students make are: (1) misreading those emotions by applying human body language to them, as I describe above. Each species has a different body language and you have to learn to become multilingual, so to speak. (2) assuming that if an animal has emotions, that it also has advanced intelligence. The two often do not go together. Some highly emotional, very social species are not necessarily that intelligent. (Canada geese again come to mind.) And some highly intelligent species have emotional lives very alien to ours due to have such extremely different social systems.