Look bro surname comes from sire. Sire is his father minato making him technically a namikaze. But im sure his kohona certificate of birth says uzimaki. So legally he is uzimaki. Now generically he is both also throw out the hair logic. Nagato was never refered to as nagato uzumaki. Look at the pic of his parents his father has black hair thats shaped like his. His mother has red hair. Likely his mother was a full uzumaki who married someone who is not.
We can go a step futher and say how do we know sasuke mother is a born into the uchiha just because her hair is black? No we havent seen her use any fire jutsu or the sharigan as hints. So we cant just assume.
But we have seen evidence of that uzumaki longevity in naruto.
