It’s never okay to post photos of someone else’s children on the internet in an identifiable way without permission. However, horses and other belongings are a different matter. Obviously a trainer shouldn’t be annoying their clients by failing to respect their wishes regarding privacy, but it is a very different thing to post photos of a horse vs. a child. Rules of thumb for trainers: only post flattering photos/comments, don’t list anything numerical/value related, and avoid listing too many specific details about the horse.
Again, I would urge the OP (and others) to try to find a way to be generous in this type of scenario. No horse makes it to the top with a magical flash of fire and smoke–there are hours, days and years of planning, worry, and hard work (typically starting even before the horse was conceived) usually on the part of many people. It is just a little bit selfish to not allow those who have been part of that arduous process to celebrate and share the success as well.
And here’s just a little piece of philosophy on the matter: horses in general are prone to accident, lameness, or simply mediocrity. When you have a horse that is succeeding, whether it is doing well in training, winning at small shows, or winning at big shows, or winning races–it’s time to celebrate and enjoy it and let others enjoy it with you.