Or maybe Anakin abandoned it at some point before he became Vader. Whatever, I am pretty sure that it WAS Anakin's, and obviously the didn't LITERALLY give it to Obi-Wan to give to Luke but the point remains that Anakin WOULD have wanted Obi-Wan to do that.
Not that Vader seemed to mind when he sent it falling down the chute at Bespin, mind.
Thinking on further what Ush said about Vader not minding when the lightsabre when falling down the chute at Bespin. Maybe when Anakin starts to turn to the dark side he creates a new Lightsabre, because a lightsabre is supposed to hold part of the characteristics of the Jedi themselves, in the changes that come about Anakin he may feel it necessary to build the new sabre, he then gives the old Sabre to Obi-Wan for him to pass onto his unborn child, I think that Vader may know that Padme is pregnant because he is quite shocked when Luke unwittingly lets his guard down and lets his emotions reveal he has a twin yet at no point is he found to be shocked at having a son.