As an amateur horseman, I'd say absolutely not. A mounted melee fighter is doing their best to deliver 5,000 lbs of force into a target, from the horse thru the saddle, into the rider's hips, torso, and out through the shoulder. It takes precise positioning and communication with the horse. But then, I wouldn't say I have the Ride feat or an animal companion horse like Xena: Warrior Princess and Argo: Warrior Palamino.
The rules of PF2e have a Tiny bit of wiggle room: Special Rules describes Tiny PCs riding along with another PC. It says,
If a Tiny PC rides along with another PC or similar non-minion
intelligent creature, roll both their initiatives and use the lower of
the two results. The two PCs act in either order on the same
initiative count. While traveling in this way, the PCs each gain two
actions at the start of their turns, instead of three, since the
larger PC spends one action keeping the smaller one balanced on their
back, and the Tiny PC spends one action maintaining their grip.
That's a Tiny PC riding a medium PC, no horse. So can a second medium PC ride a large animal companion? Assuming the encumbrance is not too much for horsey, here's how you might try to allow it:
Command An Animal or Ride feat?
You're riding the wrong way. The animal companion relationship and the Ride feat relationship don't apply to this kind of riding, so the rider must still 'Command An Animal', rolling Nature against the animal's Will DC. The animal's attitude toward BOTH riders and temperamental adaptability toward unusual situations will matter a lot. As such, a Savage vs. Nimble animal companion might have an appropriate bonus or penalty.
Failure or Critical Failure
Failure may cause the animal to Buck with a Reflex saving throw to stay on and not land prone. The second rider, without a saddle and mounted on the hindquarters, will have a result one step worse than usual. Success->failure, and failure->crit fail.
Roll twice, take lowest init
That part seems pretty clear. Roll twice and take the slower initiative.
Hold on for dear life
Without a saddle or a favorable position ahead of the horse's hips, it's like riding a half-ton pogo stick. We can extrapolate from the rules for a Tiny PC:
- For every Command an Animal action the main rider makes, they must spend an
interact action to balance the second rider.
- For every Move the horse makes, the second rider has to spend an Interact action holding tight to the main rider.
That means it's risky for transport and approach, except with a skilled rider and a horse that's not too temperamental. Then you get a double move and the main rider gets 1 more action. That's it. So the melee fighter would do best to drop you off on the edge of the battlefield.
Valets can have their own horses
Lastly, in real life, mounted combat requires a valet to constantly hand you things. Pathfinder has paid lipservice to this as well. Look up the familiar ability, Valet, which implies that a hireling could hand you semi-bulky things during combat to preserve your economy of action. This is a decent reason to take the Hireling Manager feat, so you can just shell out 1g/day to make it work.
Handling a horse and lance in combat is a pretty busy economy of action. Just watch some jousting on youtube for a minute:
Some of the competition in jousting is your pit crew. If your valet is slow to rearm you, your opponent gets a few more steps in their charge. The most common scenario for unhorsing someone is that they broke their lance on your shoulder in the last bout, and it took them an extra 2 seconds for their valet to hand them a new one, giving you extra space to charge.