Anakin and his mom were slaves, thus no one really tried to learn their last names. After the first movie, Anakin left Tatooine and his mother, Shmi, was bought from Watto by Cliegg Lars, making her Shmi Lars, step mother to Owen Lars, Luke's uncle.
This then left only a few people to know the Skywalker name, other than Ani leaving to become a Jedi. Watto eventually passed away, the Jedi were nearly completely wiped out, Luke went to live with his grandpa, uncle and new aunt.
So, by the time Luke grew up, and the setting / time line for Episodes IV, V and VI, the Skywalker name was unknown as those that knew it were either too old and may have forgotten about it, died, or had left Tatooine all together.
Besides, as Obi Wan said in Jedi, "Your father was seduced by the dark side of the Force. He ceased to be Anakin Skywalker and became Darth Vader. When that happened, the good man who was your father was destroyed. So what I told you was true... from a certain point of view." So with people believing Anakin had died, there was little or no connection made to those on Tatooine with Luke and Anakin having the same last name.
Just my 2 cents.