As we all know, the Vortex clan is a very powerful clan in the early stage of Naruto, and everyone's strength is very superior in the whole clan, so Namikaze Minato was able to marry Uzumaki Kushina because Namikaze Minato was adopted to the Vortex clan, so he named his child Vortex. Although the Vortex Team was very fierce in the early stage, it was a very powerful team in the whole world, but in the later stage, the personnel became more and more scarce, so Naruto Uzumaki was named to keep the blood of the Vortex Team.
Giving Naruto such a name is essentially to protect him. People in the village hate Naruto Uzumaki because everyone knows that there is a nine-tailed fox hidden in Naruto Uzumaki's body, and it is precisely because of the existence of the nine-tailed fox that many people's lives in Muye Village have undergone earth-shaking changes. If Naruto knew from an early age that his father was Namikaze Minato, and his body contained the power of the nine-tailed fox, he would probably be betrayed from an early age because of the ridicule and supercilious look of others.
Therefore, there are two main reasons why Naruto Usagi doesn't share the same surname as Namikaze Minato. Namikaze Minato and Vortex nine planets Nye appear in the whole animation for a short time, while the scenes of Namikaze Minato and Uzumaki Kushina's long death can only stay in the reincarnation of the soil and the memories of past lives.