Because all of Minato's enemies would have been after Naruto so the 3rd Hokage gave him Kushina's last name. Also Nagato was the most suited because he was an Uzumaki, distant relative to the Senju which means he has some of the Sage younger son DNA in him. Remember at that time the Uzumaki clan was wiped out except for a few that scattered so Nagato was probably the only one they could find. Mito was dead, kushina had the 9 tails (kushina would died after Tobi takes the fox from her making the Rinnegan useless) and more importantly Nagato was a kid and at the age where he can be mold into the weapon Tobi wanted.
ok i know that , but that doesn't make him a uzumaki , you don't get it don't you
i'm from the alashkham clan , my mom is from a sefaw clan , in my country
it is literally impossible to count me as a a sefaw , even if my father was unkown i will be called ahmed sefaw , but i will still not be a sefaw
if the father is namikaze than he is not a uzumaki , even if he is called so , it's how clans work