The light saber bit always felt made up by Obi-wan to me. No self-respecting Sith would use anything but a red crystal lightsaber. Vader needed a new one and discarded his old lightsaber.
The old lightsaber became a symbol of all that Anakin was and could have been. But I think it had more worth to Obi-wan and Luke, than it had for Vader. It seems to me he didn’t so much as pass it on as throw his former friend and master’s good intentions in his face.