It is really hard to actually describe what this means, though. What's the actual distinction between "they so strongly identify with the animal" and "they are the animal, trapped in a human body"? Is it just the desire to tell us "ok but you know you really are human, right?" I know my body is a human body, I know my DNA is human DNA, but that doesn't say anything about the mind inside this body, the concept of self. What I experience can best be described by a thought experiment: imagine that, through some magic spell, an animal was put into the body of a human and had to learn how to fit in to human society. What would that animal be feeling after all that? Probably the same way I feel.
Ultimately though, it doesn't really matter how one tries to describe this. None of the descriptions will be sufficiently accurate. But what matters is that treating me as the animal whose identity I claim makes me happy, and it doesn't harm anyone.