Look, I get what you're saying, but humans actually have the ability to say what they mean. When they can't or don't, many times it's on them.
Too often animals are not given that opportunity. Particularly around the end of their lives when humans are too selfish to realize that it's time to let them pass on.
So, yes... "emotional sensitivity," but really, it's the "for animals" part that more people lack than one might want to accept.