As it has been shown that Naruto's new power is of kyuubi, but I don't think it is kyuubi's chakra even if it is it is his chakra in the purified form, and it is the chakra of SO6P, because as Kyuubi is the strongest among the bijuus the SO6p had to put a lot of his own chakra in the kyuubi and it might be possible that he made it so that whoever can control the kyuubi they can gain so6p's chakra as well, if u ask why didn't kushina or Mito(1st's wife) didn't get it, then because they didn't have the control over the Kyuubi