Diogenesister:
Interestingly enough, in Bible times a Jewish identity was reckoned from the male line and only after the founding of Rabbinic Judaism did it change to the maternal line...
Actually it was likely in the time of Ezra, which was in Biblical times, that matrilineality in Judaism, was likely set, around 460 BCE, due to Jewish men taking on foreign wives. But it could have been set as late as the period of the Tannaim (10-70 CE) who were Jewish sages contemporary with Jesus and the Apostles that recorded the Mishnah.
Either way, it was likely set prior to the so-called "Rabbinical" period that Watchtower teaches JWs to look down upon so heavily. The reason why it is usually attributed to Ezra (at least traditionally so) is because of the Torah's narrative of the Patriarchs and the Book of Ruth (which is believed to come from the time of Ezra as well).
For instance: Abraham's son Isaac is Jewish because Sarah is Abraham's half-sister, who according to tradition, makes her Jewish. But Abraham's other son, Ishmael is not a Jew because his mother, Hagar is not Jewish--get it? Issac had two Jewish sons: Jacob and Esau. Jacob's sons were Jewish because he married wives who were Jewish relatives but Esau's son's were not Jewish because his wives were Canaanites. Get the picture? And it goes on and on throughout the Torah this way.
You are Jewish, but only if your mother is Jewish "according to what is written in the Torah," because it is just not a book of stories to Jews, but because this it is the Mosaic Law we are taking about.
In the Book of Ruth, the Moabite Ruth converts to worship the God of Abraham, making her a Jew. So when she is married by a "redeemer," she legally becomes the Jewish mother of Obed, the forefather of King David. Why? Because your mother makes you a Jew. This is even mentioned in the geneologies of Jesus.
In fact, the apostle Paul says Gentiles become spiritual Jews, or a spiritual "son" through the martilineality clause at Galatians 4:4-5 due to the fact that Jesus was born via Mary:
But when the set time had fully come, God sent his Son, born of a woman, born under the law, to redeem those under the law, that we might receive adoption to sonship.
So if you are a Christian and believe you have a spiritual "sonship" it is because of this ancient belief system in Judaism that comes from being "born of a worman, born under the law." That is what Paul was talking about.