@Monayaki They can ride horses belonging to another Sim.
So the way that I've been doing it is to go into Manage Households, use Play with Genetics to add a family member for my Sim plus horses and create a separate household for them. My current Sim, Keiko in Mt Komorebi has no room for horses but her brother Jae has the stables at his home in Britechester. Keiko can visit anytime she wants to and go horse riding. Another Sim, Ella from Chestnut Ridge is a horse trainer with Stables and she goes to her parents house in Newcrest and takes their elder Clydesdale out for a trot when they're at work and because they don't have riding skill. He's a retired military horse in their storyline and she helps care for him.
But it doesn't need to even be a family member. You could create a Sim with a few horses and a stable (different to the main competition one) where Sims can visit and take the horses out. You could even befriend Sienna in Chestnut Ridge and take Flapjack out, for example. One of my Sims is her daughter and she visits and takes Flapjack out all the time.
But it sounds like you'd like them to be family horses, so maybe use the first option and create some family with stables for your Sim to visit.
Edit: I've tested adding a second residence today using Get to Work Retail Lots because I often use one as a weekend home or add a flat or adjacent residence where they stay when working at their business. I wondered if I could create a stable at a retail lot so that it would be possible for the horses to sleep there. But sadly it doesn't work. As soon as the Sim leaves the retail lot, all pets follow them home. I'd been thinking a little store with stables where the Sim could work during the day and then leave the horses over night would be ideal as I know someone local does that IRL.
But it wasn't to be 😔.
