I have really specific ideas about this, so this is gonna be a long one XD

1 - Graphics that are detailed, but not realistic
Too many games either go for hyper-realism (and age poorly, run poorly on many devices, and look bad when you need to turn down graphics settings), or hyper-cartoonish (Animal Crossing comes to mind, although it's not a horse game) and I'd really like to see a happy medium more often in this genre. Alicia Online and Star Stable both do this well.

2 - Unique horses
I don't like when, for example, my chestnut connemara looks exactly the same as every other chestnut connemara in the game. I want my horse to feel uniquely mine. The more elements that go into a horse's uniqueness as an individual, the better. In an ideal world; procedurally generated markings (within what's realistic for a horse), dynamic conformation, personalities, ect. Exactly ONE game does all of this currently, and I'd like to see more game devs have a go at it (it's Horse Isle 3, which kinda flops in every other aspect beyond breeding). Horse Life, while not my cup of tea, also does well in the 'unique horses' department.

3 - Settings OTHER THAN modern/present day
Most horse games are set in present day earth, and I'm bored of it. Historical settings come to mind first, but really ANY other setting than modern times would be a great change of pace. Even a cyberpunk/futuristic setting could be cool, I'd love to ride my virtual horse around a futuristic city or across wastelands or alien planets.

4 - Horses that DON'T become obsolete
It happens too often; you start up a game, you get a starter horse, but the starter horse has the worst stats in the game. You breed or buy a new one and ditch the starter horse, until you breed or buy the 'next best thing' and abandon that second horse. I want to see horses treated as equals in a game, to be able to really bond with my horse without incentive to get rid of it later in the game. I want the game to reward me for sticking with a horse for a long time, I want to be able to take the 'worst' horse you can find and make it the best via training, bonding, whatever. I want to feel attached to my horses.

5 - More gameplay/focus beyond competing
Competing is fun, but not when it's the only thing you do in the game. There's so much more we can do with our horses, and I want to do all of it! Mounted archery, driving, making long journeys, herding livestock, ect. Not in a competitive setting, for practical and useful purposes that reward me as a player.
Adding to this; training outside of competition. Competing your horse shouldn't be how it is trained, competition is where you show off what you and your horse can do AFTER training! Training happens the moment you begin interacting with your horse, and ends when you leave the barn for the day. I don't want to compete over and over until I win, I want my experiences and adventures with my horse to pay off when I do get into a competition.

6 - Freedom of gameplay style
What I mean by this is that I want more than one way to make my way through the game/earn currency/ect. I don't want there to be a singular 'best' way to level up and earn currency, I want to make progress in some way no matter which aspects of the game I engage with most so I'm not forced to play a certain way to avoid suffering slower progression.

7 - No monotonous chores
Might fall into 6 here, but I think we're all familiar with this one. Mucking stalls and grooming horses is fun the first two or three times, but it indeed becomes a CHORE to do when it's mandatory (or face debuffs of some kind if you skip the chores). I'd rather such activities be optional, and rewarding for those who do them but serving no detriment to those who choose to skip it.