If you all could recall in the Phantom menace(or watch the movie again!), Anakin skywalker, was born without a father, Guigon gin once ask his mother Shmi who was the father, and she replied that there is no one, she sort of just conceive as if the force creates the life itself.(which is quite religious, but this is science fiction anyway) That also explains why he is so desperate for a father figure, initially Guigon but unfortunately he died, and later with Palpatine.
So it seems Anakin adopts his surname from his mother Shmi skywalker. There are obviously many skywalker family around some are slaves, from Shmi's earlier descendant lines. No one is suppose to know that Luke & Leia existed, until Vader sensed a disturbance in the force when Luke started to learn to command the force. It was then confirm that indeed Padme have given birth before she died. Vader first learnt that Luke was his son and later discovered through the force that Leia was connected to Luke. I remember him saying that 'if you(Luke) will not join me perhaps your sister will'.
Just because Anakin help the emperor to dominate the galaxy and caused so much hardship, its not surprising that anyone associated skywalker name will not be welcome anywhere in the galaxy, Luke for one is unlucky and to him its the shock of his life to find out he has a direct link with anakin. ;D
Its interesting to read Dark horse comic too, you could find more detailed stories.
Nice one! I suspect it was only by speculation that many thought Anakin was Vader, only a select few knew about the connection and none of them were named Perez Hilton - not likely to blab to the cosmic gossip columns. His turning to the Dark Side was sudden, followed by the Jedi culling, and his fateful duel with Obi wan. Word wouldn't have gotten out in time.