~Closed as this is just a breeding ground for flame comments.
Naruto is an Uzumaki, at least for 50%, for the simple fact that his mother was one and that's not even debatable. No matter which name he carries; whether it be Uzumaki, Namikaze, Senju or anything else or nothing at all; in terms of blood he is related to the Uzumaki clan and by that carries blood related to the SO6P. Neither Nagato or Karin has ever referred to themselves, nor has it ever been stated they were even aware of their Uzumaki lineage, as Uzumaki's still both Kabuto and Obito named them like that because they have Uzumaki blood, just like Naruto.
That whole naming business is quite irrelevant. The only point it has some value is to decide whether he officially belongs to the clan or not, but it's pretty useless to start nitpicking on that as there is no 'official' Uzumaki clan anymore as far as we now as their village got destroyed and its former members spread all over the place in different villages nor has there been any indication that something like a Namikaze clan exists. It's not because it's a surname it means it's a clan.
So yeah I really don't see on what now you are exactly picking on as by blood he is an Uzumaki. Anyone who claims otherwise is or ignorant or is trolling as it's a manga fact, he has their blood. Considering the Uzumaki clan doesn't exist anymore as an organized group and there is no proof that Namikaze is a clan, I really don't see what now your problem is whether he carries the name Uzumaki or Namikaze, he has SO6P blood. On top of that I am not really knowledgable about how people get their names in Japan, let alone if there is a certain system in Naruto, however I do know that there are places where it's normal to get the surname of your mother. So nitpicking just on his name is not needed as that's nothing more than an artificial tradition.