Once a horse is sold, it is sold, just like any other property. All a breeder can do is produce the very best animals they can, so the horse will always have a job, and likely remain sound, being in the care of people who can recognise a good horse. And register the horse in a recognised stud book because future buyers like papers (even, I believe, those who speak of “ride the horse not the papers” when given the choice) and people will often pay more for a pedigree animal and, finally, many stud books maintain performance data that can also offer some protection to closely related horses.
Then the breeders just have to hope that the name of the horse isn’t changed somewhere down the line. Certain names stick, however, especially if they have some kudus because they offer status to the owner e.g. an Event rider isn’t going to show “Fluffy no name” when “Cooley Fluffy” or a Dressage rider when “Mount StJohn Fluffy” indicates or implies so much more… Or even a QH stallion called “Bet Hesa Cat” is a bit of a giveaway too.
In Spain and Portugal the studfarm brand is often the first point in a description and at shows “the cobra” is a group of mares shown together to demonstrate the quality and depth of the breeding mares.