We shouldn't anthropomorphize animals. Instead, the point here is that emotions are not anthropic; humans do not have a monopoly on emotions.
Which is to say, we should be sure not to think that "grief" means the same thing for an orca that it means to a human; yet naturally animals do grieve in their own ways. (Heck, "grief" doesn't even mean the same thing for all humans!)
in other words, all the ways we have to manage emotions, and all of our "non-emotional" functioning.
So we have somehow evolved to be able to better manage emotions.
Ok, so if we grant all of that, then haven't we just said that emotions are common to animals? So then how is it anthropomorphising to say that animals have the traits which we've just said define animals, whereas what makes humans special is that we have 'risen above' mere emotion?
Insects don't have emotions, they barely have hyper specialized sensors as brains.
However, emotions efficiently direct behaviours in mammals. They enact immediate and persistent responses.
We can say without much doubt that emotions in mammals are similar to our own emotions.
The big unknowns are animals further away from us, which are demonstrably intelligent, like birds and octopuses.
What does REM sleep have to do with emotions? REM sleep is retraining of the brain over the experiences of the previous day. It is a pure intellectual activity, it doesn't need emotions.
In fact, emotions are not an emergent feature of brains just by themselves. They require parts of the brain specifically dedicated to emotions. Like a limbic system.
As I said, Octopuses can have something like a limbic system, with their big complex brains.