Can I Keep a Stalhorse?

Link can try to mount them, they will automatically be tamed. He can ride one just like a living horse, but like wild animals it cannot be registered at a stable. The Stable Clerk will indeed consider it as a monster and refuses to register or board it out of fear that it may eat the living horses.

Can you own a Stalhorse?

Unfortunately, you aren't able to register a stalhorse in BotW. Upon trying to register the skeleton horse, as 64Jolink on Youtube points out, the stable owner is shocked to see you're riding "a monster, not a horse." Above all else, he's worried it'll eat the other horses.

Can you save a Stalhorse?

Characteristics. This skeletal horse is ridden by monsters. It was once a regular horse, but Ganon's power revived it from death. It cannot maintain its bone structure in the daytime.