Riding on a different horse will make that horse the one you own. What happens to your previous horse I do not know. Suggest you load a previous save if possible.
Oh, so if I'm sick of that horse I picked up at the Fort (which is lovely, nice, helpful and everything, but wanders back to the Fort every time we reach our destination, only to magically appear every time I fast travel to a new destination), all I have to do is formally buy my own horse, hop on it, and the old loyal horse will just wander back to his old Fort permanently and be safe there?
Will that also work if I got Dawnguard and rode the summoned horse?