Even the BO, who is the director of the rescue and therefore the owner calls him "my horse or Castles horse" she's always hitting me up saying "your horse misses you, your horse wants you to ride, your horse.....etc"
I'm always quick to clarify that I DO NOT own him if people inquire.
I think people biggest worry over claiming "ownership" is that you mat feel one day that you are entitled to the horse and then try to play the system and take the horse....
Does that make sense??
IMO its pretty harmless as log as you aren't trying to make the wrong impression
But if you clarify and you don't mean anything shady by it, then I never see a problem with it.