So, I've just seen the move once, but here is my recollection ...
Kylo clearly has the ability to read minds from the scenes we saw in the film. He also seemingly realized the internal conflict going on within Finn when he saw that Finn just stood in one spot during the battle and did not follow orders to kill everyone. Now, what we don't know with that particular force power in this new universe is what can be discerned from others' minds at close vs far range. So, if Kylo could sense the conflict within Finn from where he was standing, then it is reasonable to assume he could read Finn's surface thoughts too.
Another possibility is that it is reasonable to assume the troopers and commanders have battlefield awareness of each other through HUDs, because why wouldn't you with the kind of technology they have? However, we didn't see this in the film; though, there was a lot we didn't see in the film.
Finally, Kylo's taking note of Finn didn't even require his force abilities. The troopers, at least Finn, seemed to be brainwashed and should not have this kind of conflict at following orders. Furthermore, there is the incident back on board the ship with Phasma, where any competent commander would sense that Finn had a problem, which she did. It is not a stretch that Capt. Phasma and Kylo would likely have discussed the incident(s) concerning Finn, which is where Kylo could have learned his designation for sure. As a commander of troops, it should be Phasma's responsibility to bring these potential issues up, especially to the powerful mind reading Kylo. Both Capt. Phasma and Kylo would have suspected they had a troubled storm trooper in their midst, and so it is no small leap to assume that when a storm trooper goes rogue and steals a TIE, it was likely FN-2187.