I always presumed that Anakin Skywalker died (figuratively) and Vader was reborn from Skywalker and that was information that only a small circle of people both Jedi and Empire were privy to. Anakin might have been a hero of the Clone Wars but the Jedi were disgraced and obliterated before the wars ended, therefore I doubt 'Anakin Skywalker' was a household name, at least not enough for the galaxy to question Luke's surname, especially as Tatooine was in the outer rim with little to no Imperial influence.

which all brings about a bigger question, is the Galaxy aware they under Sith rule?