This isn't a compete answer since the canon doesn't have an answer yet.
How did he recognize? Not through Force!
However, your second guess (some special Force signature) is not correct. According to both WGA leaked script, and Alan Dean Foster's TFA novelization, he recognized the lightsaber on sight:
In reply, Finn TURNS ON LUKE'S LIGHTSABER -- upon seeing the weapon, Kylo Ren REACTS IN SHOCK.
KYLO REN (CONT'D)
That lightsaber. It belongs to me!
He pulled and activated the Skywalker lightsaber.
For some reason, the sight of it was enough to give Ren pause. He stared at it for a moment before reacting.
“That weapon—is mine.”
Finn all but snarled his reply. “Come and get it.”
Please note that he is unlikely to have sensed it through the Force, because he ONLY reacts to it when Finn unleashes it. As opposed to, when Finn and Rey are above him while he deals with Han; or when he first confronts them in the forest. For that matter, if he Force-sensed it, he'd have turned back and gotten it from Maz Kanata back on Takodana - he wasn't that far from the castle since Finn saw him carry her to the shuttle.
So how did he know?
In TFA novelization, Leader Snoke tells Kylo Ren that he witnessed rise and fall of the Galactic empire. And he tells him details of what happened on Second Death Star between Luke, Emperor and Vader.
It is not impossible that he also told Ren what happened in Cloud City.
in Aftermath book, we have a group that seeks to buy Vader's lightsaber from a dealer. That means the information about such things is publicly available in general.