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The bloodlines were obviously not that distant. Even Hashirama married Mito Uzumaki.
Senju do not have any distinguishing features to tell them apart like the sharingan or byakugan so it was inevitable that the clan would have mixed into other clans and dropped their last names. Unless you think the Uchiha interbreed, it's the same thing with them, except that they have their sharingan.
Naruto and Nagato are confirmed in canon to be of the Senju bloodline. This isn't up for debate.
Uchiha interbreeding is irrelevant. Hyuga are related to Uchiha as well, that doesn't mean Neji can be called an Uchiha. Naruto is an Uzumaki, that's that.