Okay so here's a bunch of thoughts in no particular order:
This is not only an in game-feature, there is plenty of art of Roman cavalrymen with a shield and a spear, possibly a Doru or a hasta (2-2.3 meters long). There is also art of them carrying Lances.
Conceptually it might help to think of how similar the foot fighting and the mounted fighting techniques were - and in some ways they were quite similar. There are actual manuals featuring wrestling and dagger moves that were presumably useful for renaissance horsemen. Now that we're thinking that way, are there any examples or infantry with a long spear and a secondary spear? Well, plenty. Pre-Marian Roman infantry, Germanic infantry into the Migration era, various elite Hellenistic light troops like the Silver Shields. One is the primary fighting spear, the other can be thrown or used instead of the primary for close quarters fighting. A secondary spear isn't much bigger than a specialised throwing dart, anyway.
And on horse you can carry more things anyhow. So there's that.
And I read some accounts in the High Middle Ages of Normans carrying Lance and a spear into combat (redundant...).
In the high middle ages and much later (into the early 1500s) in Spain, secondary spears were often thrown by hand (no cord like in classical times). So there's that again. The second part is HOW long poking things can be used on horseback.
What's the use of the one-handed spear for cavalrymen?
What use is a one-handed spear to an infantryman? Same here. You can block, parry, and counter with it very easily. It outranges swords. It is very very very very hard to stop when it's coming directly for your face or whatnot. It defends you from enemies riding up behind. Horses are much nimbler than people think. They turn on a spot and sidestep in any direction unless they have a lot of momentim. If ridiculously well-trained, they will outmaneuvre people on foot. The men on the horses are also fairly nimble and there aren't that many blind spots which they cannot reach with a sword (immediately in front of the horse and under the horse's belly, mainly). A one-handed spear would let you reach a man trying to escape you by lying prone (underhand) or stepping in front of your horse to scare it (overhand, same as lots of polaxe blocks).
There are two basic ways of using a lance or a spear on horse: one is to couch/brace, the other is to hold it in hand and then stab. Well, even the heaviest "jousting" lances have plenty of stabbing techniques illustrated in Renaissance fechtbuchs. You can hold one in one hand overarm, brace it over the opposite forearm to attack sideways, sweep it around defensively, angle it to strike with it on the "wrong" side of your horse's head, strike behind yourself, etc. So basically exactly what you get in manuals for 19th c. lancers who had to learn way more tricks their weapon could do than a cavalry sword, plus a little more.
A spear just does the first thing (couching) worse, and the second thing (fencing at close range) better. It's arguably better as a secondary weapon than a sword. And you can absolutely kill people with a thrust from a short spear. In fact, you can split skulls with a spear-blade just like with any cavalry axe if you swing hard/accurately enough. The spear is highly versatile, people knew how to use it in a variety of ways, and the surviving instructional literature reflects that.
A one-handed one allows the free hand to quickly use a shield you strapped to your back I suppose.
I can't vouch for every horseman in history, but I imagine it rested on a shoulder-sling just like lances did, or hung off the saddle when not in use.
Also, not every shield is the same. The Farousiyya (a 14th c. manual) talks a lot about fighting with both short spear and lance (mostly in terms of fencing rather than couching), and also assumes the rider would have a smallish shield, to protect the fighting hand (whether he is a lancer or an archer). This is exactly the same as the small shields worn by phalangites, or bucklers worn by infantry (or indeed horsemen all over the Eurasian steppe into the 19th c.). It offers active protection against melee weapons mostly wielded by other horsemen, combined with the primary weapon (the lance or spear used one-handed). If it saves your wrist from an arrow, that's just gravy. If it deflects a footman's spear attack, even better.
A bigger shield like the Normans used would presumably be more useful against couched lances or maybe arrows (defending the rider primarily). Once again I think the primary opponent was another horseman though I heard the long kite shape protected the leg on the opposite side of the sword-arm, so that could be meant to deal with infantry at close range.
I personally think the decline of the secondary spear probably has everything to do with armour becoming cheaper and better. Fencing with a short spear against a well-defended opponent is likely ineffectual. Instead you get tabar-style axes in Iran, maces/picks in Europe, konchars/estocs in the Renaissance, and ultimately pistols everywhere. The sword was retained because it weighs very little, can be worn without much burden, and is still useful at killing unarmoured infantry in pursuit.