The Mongols conquered nearly all of Eurasia on horseback, while shooting arrows from their composite bows. All pictures of archers on horseback clearly show they do NOT hold on to the reins. Riding without holding on to the horse does not seem to be a major issue.
However, all pictures that I can find of knights wielding lances on horseback shows that these definitely DO hold the reins in the hand with the shield. I think that it is probably essential to hold on to something as your lance slams into the enemy.
I was a professional stunt jouster for 5 or 6 years, and although my experiences were never battlefield tested, they should still count for something. That being said, I'm torn over whether real life alone should really dictate whether or not something like this should be allowed by a DM.
My entire schtick was a Mongol warrior riding around bareback and doing spear and archery tricks. Using the reins is the easiest way to control the horse. If you're not using the reins, you're directing the horse with your knees and your feet and relying way more on the horse's training.
When you're jousting (or really when you're doing anything where you're able to hold the reins) you're using those reins to tell the horse what you want it to do. It's your steering wheel. They're not helping you stay on the horse, they're just there so that you can convince the horse to (hopefully) go where you want it to go. What's keeping you on the horse are your legs and your saddle, and you leaning properly into the hit that you took and the hit that you landed.
In D&D two weapons mostly translates into more attacks, but in real life two weapons means more opportunities to take advantage of an opening to
land an attack. I haven't
jousted with two lances, but I've gone through the motions of a staged fight with various combinations of two light lances and/or spears and it's totally doable. Is it realistic, I don't know. You can at least be cinematic with light lances and spears, you've got backup weapons, you can throw them, they're small enough to spin and block things. But heavy lances? I don't see it being very practical, I don't think I ever had the strength to be able to do anything with one that wasn't just couching it and making a pass with one. Jousting with two heavy lances? The horse is in the way, the lances are in the way, unless the thing you're hitting is really big you're constantly fighting with the lances effectively being different sizes. I wouldn't ever want to do it.