Being a breeder mean that you make birth a foal.
At the beginning you're at the same time the breeder and the owner.
When the breeder sells his foal, he gives up ownership to the customer.
He's no more longer owner but he still is the breeder's foal, he's the one who gives it birth !
It's important to notice it cause, when a foal becomes a horse and wins prizes in competition, a part of these prizes comes back to the breeder.
To summarize : A horse can have a lot of different owners in its life but the person who gives it birth (the breeder) never change :)