I suspect that where this gets strange, is it may actually be a sign of something biologically broken in us. Humans relate to other species, bond with them, interact with them readily, etc, because the innate self/non-self instruction set many mammals have doesn't work in us. You can see this in action with dogs as opposed to wolves, btw. Wolves won't acknowledge a human socially, unless that human howls at them, which is apparently part of their integral self-species recognition. Dogs, however, have been bred so that little feature doesn't work anymore, and they see you as just another dog, albeit a rather odd, incredibly large one.
This little software bug is, I hypothesize, related to all manner of quirks of human behavior, one favorite example being furries. For one thing, we used to be a hell of a lot fuzzier, for another, we have no innate sense of what an appropriate mate looks like, and since we're wired pretty similarly to Bonobos, IE having a tendency to use various flavors of getting one's freak on, as a social lubricant... well.. you add non-human animals in affectionate relationships to this mix, and some of us are going to get a bit.. buggy.
TL;DR, I suspect humans empathize with other species due to a software error, one of those few cases where a bork somewhere in the system actually made the product function waaay better, but not without some quirks.