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Zetsu doesn't say that Nagato is a Senju, but that he's from their lineage. We've already been told that the Uzumaki and the Senji are distant relatives, so any and all Uzumaki are by definition from the 'Senju Lineage'. Also, even though red hair is a trademark for the Uzumaki clan, that doesn't mean that all of them have red hair without exception. Just look at Naruto. There's also the possibility that the anime producers simply did their own thing when colouring the hair.
Pretty much with Azuma here, the Uzumaki is a branch off the Senju.
Perhaps they realized that they could not keep all the traits of the great Sage in their bloodlines, and needed to protect them, thereby they did split the Senju into Uzumaki and the remaining staying as Senju.
It's possible that the Uzumaki having the Great sage's stamina and more sealing technqiues etc. had to become a clan on their own, in order to preserve this quality.
It's likely that sometimes a strong Senju and a strong Uzumaki were to be married though, like it did happened with Hirashima and his Uzumaki wife.
In this way, they can keep the traits in the 2 different clans pure, but sometimes marry strong individuals, and produce a child that is closer to the Great Sage himself, and at that, they can keep circling their bloodlines between the two clans, to keep as many traits from the great Sage as possible.
Why not just stay in the same clan one might ask?
If the Uzumaki were to stay in the Senju, then perhaps their unique bodies will start to disappaer, when Senju doesn't seem to have the same kind of unique body as the Uzumaki did enherit from the great Sage.
That's likely why they need to preserve two different clans, in order to keep their bloodline traits in control.
If someone shows up with strong Senju genetics, and another with strong Uzumaki genetics, then that is a perfect opportunity to re-build and maintain the genetics from the Great Sage into a single individual.
Well this is just speculation in the end, who knows why the Uzumaki really did split up, but we know that Senju and Uzumaki are really close and friends, so it's doubtful that the split was on bad terms, rather, it most have been of constructive reasons, like for example to preserve their bloodline from the Great Sage.
Nagato is an Uzumaki, but I think Madara was saying that Uzumaki being the sister clan of Senju is what allowed Nagato have some Senju traits and ultimately be able to control Gedo Mazou.