In our campaign, the Cleric has a 'pet' spider which he uses for his Giant Vermin spell. I (the druid) have agreed to take the Vermin Heart feat and cast Awaken on the spider when we next level up. We'll then dig out a wizard to cast Permanency to keep the spider large.
Having a giant spider as a party member will be an interesting experience. Of course, I haven't fully mentioned that Awaken means that the spider will initially be bonded with me....
In return, until the spell the Cleric has promised to look after the spider, make sure it has food and water and clean out its bedding every day. It'll probably end up with me doing it all, though, tcha.
There's absolutely no reason I can see you can't give class levels to a horse you bought. Limited to intuitive classes, probably. Said horse would also have to contribute to the combat it gains experience in, take an equal share of the experience, and count as a party member for APL calculations. Also deserves a full share of loot too, probably. If your party is cool with you playing two party members, one of whom is a horse, then that's entirely a possibility.
If you want the one-off ability to give a couple levels to something (and you're not the DM), well, I really don't see that happening. Gaining experience isn't something you can just turn off, either you're an experience gainer (PC) or you're not (everything else).
So my horse could take levels in Paladin for the free Heavy Armor proficiency and Lay on Hooves ability to mule kick the life *into* me... I like it.
Even with Enlightened/Awoken/Magical creatures it is still completely a DM's fiat to reward anything like exp or other bonuses to an Animal type.
I do see it though... Imagine that same hypothetical Griffon, defending a cave from other animals and adventurers. Does it never get stronger, faster, tougher or smarter from surviving those encounters?
He's essentially asking for the same logic to apply to one he buys. Unfortunately there is no rule that explicitly supports it, and since rules trump logic it would be up to DM.
The reason that there are no rules for any NPCs to gain XP is because they gain levels as required by the GM.
If the GM wants a 10th level Tiefling commoner to suddenly become an 11th level commoner it happens. If the GM wants a shadowdancing megafauna, paladin horse or even (gods forbid) a ninja chicken that is exactly what happens, there's absolutely no need for xp. If there is a griffon with the party it gains levels if and when the GM says so and in whatever class the GM allows unless you have a class ability or feat that says otherwise (or you somehow persuade your GM to allow you to play as an awakened chicken, but then that becomes a PC so...)
(this brings back bad memories of barbarian vampire sheep...don't ask!)