Well, although Uzumaki are a separate "parent" clan if you will on the same level as Senju and Uchiha, they do have a distance blood relation to Senju, so in that respect they are closely tied to Senju in the way you described, so while Naruto isn't directly Senju, in order to provide that inherent rivalry with Sasuke of Uchiha, he is someone who has the blood relation to Senju in order to carry out it's feud with Uchiha.
Re-reading that chapter though, I did notice something that nobody else here has mentioned. That being that Kushina was the second kyuubi jinchuuriki. Does anybody else think the specific mention of "second" was an important qualification to make? She could have just been a "former" kyuubi jinchuuriki. Who was the first? What is Madara's fascination with the kyuubi? It's the one Bijuu he's had the most contact with. Is it possible he was the first? Who else (of importance) was around back then to have been a jinchuuriki for it?