Luke Skywalker was hidden on Tatooine from the Empire in the original Star Wars, but it seemed odd that he was openly using his father's last name while doing so. The legendary Jedi Master started as a humble moisture farmer on Tatooine before he joined the Rebellion's fight against the Empire. During his involvement, he was also faced with the harsh truth that he was no random person who happened to be strong with the Force as he turned out to be the secret son of Darth Vader.
After the original trilogy ended, George Lucas expanded the universe with the prequels chronicling Anakin Skywalker's backstory. Starting as a slave living with his mother on Tatooine, he trained to be a Jedi and was once dubbed the Chosen One only to succumb to the dark side, assuming a new persona as the Sith Lord Darth Vader – Emperor Palpatine's right-hand man. When Anakin's wife, Padmé Amidala, gave birth to their twins, it was decided by Obi-Wan Kenobi, Yoda and Bail Organa that it would be better to split the kids: the girl was adopted by Organa, while the boy was taken to Tatooine and given to Owen and Beru Lars to raise. While Leia's identity was concealed by becoming an Organa, Luke kept the Skywalker name, which doesn't make much sense for someone in hiding from a father who has the same name.
This may sound like a Star Wars plot hole, but there is justification for why Luke used Darth Vader's real surname. For starters, no one really knew what happened to Anakin; there's no official reason in canon thus far that explains when and how Obi-Wan Kenobi learned about his former padawan surviving their duel on Mustafar to become Darth Vader. Given this, it's possible that when he handed off Luke to the Lars family, he was operating under the assumption that Anakin is dead. Giving his son his last name was a way to pay tribute to the former Jedi considering that they were once very close. On the contrary, since Darth Vader was unaware that Padmé gave birth before dying, he didn't have any reason to look for his children since he didn't know about them. In hindsight, Luke had the luxury of carrying his father's real last name because he was a nobody hidden in the Outer Rim Territories. Leia, on the other, was adopted into a well-known and prominent family, hence why she was given the name Organa.
In the real world, this creative decision boils down to the idea that Star Wars creator George Lucas didn't initially plan Darth Vader to be Luke's father. Instead, Darth Vader and Anakin Skywalker were meant to be two different characters, with Skywalker being a Jedi Knight among the last to be hunted down and killed by the Sith Lord. This tracks with one original Star Wars deleted scene where Biggs Darklighter referenced Luke's dad to back up the Rebellion newbie's flying skills to the Red Leader as they gear up to disarm the first Death Star. Given this, it makes sense that Lucas wasn't concerned about giving Luke the Skywalker last name in the series' first installment as it was yet to be decided that he would be the secret son of Darth Vader.
With the Skywalker Saga done, it's easy to dissect its smallest details because all its twists and turns have been laid out. Admittedly, it's fun pondering on these questions, especially considering the current lack of new Star Wars content. But as the franchise thrives moving forward, perhaps Lucasfilm can provide an official canon explanation to this particular query, potentially through the upcoming Obi-Wan Kenobi series on Disney+.
