It's basically that, minus the ability to talk (the "daemons" in HDM can talk like people, whereas the soul animals in DSA can only talk telepathically with their respective witch). The only other difference is that a witches' companion is usually born elsewhere and has to find their witch in the real world later, which usually happens when the witch is around seven years old and her magic awakens. IIRC the daemons in HDM are somehow born alongside their respective humans and keep shapeshifting until they settle for a matching animal form eventually.

I fully agree with the name change tho. I mean, it should not have been that hard to just stick to Blackfurs and Redfurs for orcs & goblins when they're "nicknamed" that way in the actual game anyway. Both races also have noticably different names for themselves in the game (goblins call themselves Suulak, for example) so that would have been yet another option for a unique name. No need to call these fictional races by a name that most people already associate with very specific traits and then go out of your way to make them "different".