Well, not strictly horses, but the ownerships do have nationalities. And for the Olympics specially, the horses' ownerships must be the same as that of their riders. But this is only for the FEI database - the real ownerships are tracked elsewhere and are commonly tweaked to make some horse-rider partnerships eligible.
What is funny in this case is that Tatiana Miloserdova is both a rider and an owner of her horse... But in one database she is listed as Italian, and in another as Russian, and therefore ineligible.
She may have forgot to change the horse's nationality, but some appeal seems possible here. It's not her fault that FEI databases are not connected.