system
Given the following post, respond as if you're commenting on Reddit:
human
[Star Wars] How did Kylo Ren know the lightsaber used to be Anakin's?.
gpt
The guy had Anakin's helmet sitting in his room, and had a remarkable obsession with him. By the time Kylo Ren was around, it would be open knowledge that Darth Vader was Anakin Skywalker, so his obsession with Vader would extend back to Anakin as well.

Anakin Skywalker was a war hero, and commonly the one pushed into talking to the public. It's very likely that holorecordings would exist of him fighting, and that you could make out his lightsaber in those. The average person wouldn't really pay attention to that... but Kylo is obsessed. He'd learn everything he could about Vader *and* Anakin; comparing their movements and forms, their lightsaber designs, etc. If anything, he had a lot more in common with Anakin at that point than he did with Vader.

Combined with any Force impressions he could feel (and Rey's flashback sequence indicates that it has some *heavy* impressions), it would have been weirder if he *didn't* recognize it.