RAW: Animal Companions can Benefit from Potions
The rules for potions state:
You can activate a potion with an Interact action as you drink it or feed it to another creature. You can feed a potion only to a creature that is within reach and willing or otherwise so helpless that it can’t resist. You usually need only one hand to consume a potion or feed it to another creature.
So, assuming your animal companion is willing (which taking food and drink from some sort from their owner seems very normal), your animal companion can consume potions.
But, as you mention in the comments, the Companion Items rules mention that:
Normally [companion items] are the only items a companion can use. Other items can qualify, at the GM's discretion, but an animal can never Activate an Item.
Good news! The potion rules also cover this; the potion giver "[activates] a potion with an Interact actions as you... feed it to another creature." So, while the potion is benefitting the animal companion, you are the one activating it and using it. If it helps, think of it like a wand; the spellcaster is the one using and activating the wand, but the effects of the usage are applied to the target of the spell. Your horse isn't using the potion; you're spending the actions and you're activating the potion, therefore you are using the item. Your horse simply benefits.
But can they benefit? As you point out, the animal companion rules state that the only item bonuses an animal companion can benefit from are for Speed and AC, with some limitation on AC. So that means potions that grant certain item bonuses, like the Gecko Potion, are useless. However, a lot of potions don't grant item bonuses - importantly, that includes the Potion of Leaping you are looking at. A benefit from an item does not mean that it is an item bonus - that's how you can use a shield with your armor!
RAW: Horses are fine to ride while Leaping
This brings us to the next question: can you ride your Horse while it Leaps? If you can, then your Potion of Leaping trick will work. Good news - your Horse has the mount special ability, so you can:
You or an ally can ride your animal companion as long as it is at least one size larger than the rider. If it is carrying a rider, the animal companion can use only its land Speed, and it can't move and Support you on the same turn. However, if your companion has the mount special ability, it's especially suited for riding and ignores both of these restrictions.
This will allow your Horse to Leap with you on it - indeed, people do this in real life quite frequently! It's very arguable (and I would agree) that you could do this with a non-mount animal companion too, but even the strictest GM should agree a Horse can. Importantly, Command an Animal tells us that most animals know how to Leap:
Most animals know the Drop Prone, Leap, Seek, Stand, Stride, and Strike basic actions.
It's not entirely clear if Animal Companions are even restricted to these actions (I'd say they aren't, since the Animal Companion rules mention they are "in place of the usual effects of Command an Animal"), but even the most restrictive GM should allow this via RAW.