Why doesn't naruto have his dad's last name?

Answer 1

Because his father thought that he have enough bad attention for having the nani tailed fox in him that is would make his life worse because people might think stuff like you killed your father or other questions like that?

Another reason would be because that Uzumaki was the clan and Kishimoto called the three of them the UZUMAKI family, NOT the NAMIKAZE....... in other words, Minoto joined the Uzumaki's, Kushina DIDN'T join the Namikaze....... Namikaze wasn't a clan, it never once stated that Namikaze was a clan, as such, Naruto is Uzumaki Naruto, NOT Namikaze

Answer 2

so nobody would know its his son