It's monarchy, rule by right of bloodlines. Nynaeve has about as much right to rule Malkier as any other foreigner, her only real authority comes from marriage to the bloodline (and being an Aes Sedai, obviously). As such, she could only rule as consort, or as regent by being the mother to the next ruler of Malkier if Lan dies before their children are old enough to reign in their own right. For the same reason, Taringail could never rule Andor (though obviously that particular country discriminates not only by bloodline but also by se nor could Perrin rule Saldaea.

EDIT: Although Cannoli may be correct about the implications of the Testing.