It was only natural for fans to question why Naruto was given Kushina Uzumaki's last name rather than Minato Namikaze's after learning the real identities of his parents. The Third Hokage, Hiruzen Sarutobi, took many precautions when charged with keeping Naruto safe, including separating him from the Namikaze clan.
Unfortunately, the majority of the Uzumaki clan was entirely destroyed, and those that did remain were in grave danger due to the strength they possessed. It would seem obvious on that basis alone for Hiruzen to give Naruto the surname Namikaze, but the Third Hokage's choice was based on logic that had nothing to do with Minato enclosing the nine-tailed fox inside his unborn child.
Naruto's life was always at risk, but he was safer as an Uzumaki than as a Namikaze.
During the Third Shinobi War, before Naruto’s birth, Minato was notoriously feared as the Yellow Flash, a deadly ninja who single-handedly killed 1,000 shinobi in an instant. This act won the war for Konoha and forced other nations to accept a peace treaty. During that period, Minato made several enemies willing to take any risk for an opportunity to carry out revenge against him and his family.
During the nine-tailed fox attack that occurred shortly after Naruto's birth, Minato, the Fourth Hokage, and Kushina gave their lives to save Konoha and their child, Naruto. Half of the beast was sealed inside a newborn Naruto, and both parents died as a result. The care of Naruto was entrusted to the Third Hokage, who, interestingly, gave him the surname Uzumaki, but why did he do so? Everyone in the village (except Naruto) was well aware of his origins, but no one ever mentioned it, especially to Naruto.
Naruto's Surname Uzumaki Was More Secure than Namikaze
After the Uzumaki clan massacre, the few survivors were hunted down and captured by those who wanted their powers for themselves, like Orochimaru. However, that hunting gradually petered out over time. Considering the sheer number of Minato's enemies, and how ninjas from other villages were intent on killing any of his relatives, logically, it was much safer to name Naruto Uzumaki than Namikaze.
Given the associated dangers, neither Uzumaki nor Namikaze were ideal surname choices. To make matters worse, Naruto was never placed for adoption because everyone in the village despised him. They knew he was harboring the tailed beast and blamed him for the death of their beloved Fourth Hokage. While the Third Hokage may have tried to protect Naruto, he was clearly not looking after him beyond keeping him safe from his parents' enemies.
Aside from the massacre, there was no recorded incident of an Uzumaki being hunted down, so the name was safer to take, and the Third Hokage never revealed Naruto's relationship with Minato (although Naruto did later find out about his parentage). This was the only way Naruto could be protected from his father's enemies. When Naruto eventually discovered the truth about his parents, he confidently told everyone that he was strong enough to take on anyone who dared to confront him for vengeance, despite the risks. He was safe from the dangers associated with surnames by that point because he could defend himself.
