He says it belongs to him because he’s a Vader fanboy and believes that his grandfather’s blade is his birthright.
I’m not even really interested in Kylo’s interest in the saber, rather I am trying to find a way to reduce the huge number of coincidences in the film. In the first 30-40 minutes, I was able to accept the coincidences, but by the halfway point the story really shouldn’t be relying on these narrative cheats.