If you want to permanently make the horse yours, all you have to do is put your saddle on it after you break it. That instantly puts you at level 1 bonding with the horse, which makes it as much yours as any horse you would get at the stables. When I go for an Arabian, as soon as I break the horse, I put my saddle on it. That way if it does throw me due to a predator I don't lose it forever, I can just call it, and it will come running back to me. At the moment, I have 3 horses. The white Arabian, the red chestnut Arabian, and the perlino Andalusian. In case you're wondering what the hell perlino is, here's a pic of a perlino Andalusian.