Pretty sure that when it comes to surname of Naruto- according to explanations here and there, identity of his parents were kept secret to protect Naruto, that's why he didn't get Namikaze surname and later wasn't told who his parents, especially father, were.

But yeah- hole in it is that people knew that wife of Hokage was Uzumaki- and suddenly a kid named like that appear- and Kushina she was pregnant... so it doesn't take much of thinking to get to fact that he's son of them.

But wasn't it like that- all Konoha's people who knew about all this- were told by 3rd Hokage to never talk about it, never mention who naruto actualy is- not just about tailed beast but also his parents and surname things and stuff. And if he would be given name Namikaze- then people even outside village would know who's son he is and Uzumaki wasn't such greatly remembered name- 4th Hokage was known as hero but who knew Kushina actualy? People from Konoha, yeah.

But also... we're talking about such details, while it's a world where some dude is still alive being chooped to pieces- actualy a talking head- burried under forest... author confirmed that Hidan is still alive btw e.e'

Anyway- it's to hot to think. Let's get some cold drinks and enjoy series and game anyway u.u