If only Noah's family survived the flood, and we are all related from them, why are we not Jewish?

Noah was not Jewish and, according to the Torah, lived long before the Jews existed. According to the story, he had 3 sons, Shem, Ham, and Japheth. The Jews form only one tiny branch of the descendants of Shem.

According to the Torah, Abraham was a descendant of Shem and Abraham fathered both Ishmael, progenitor of the Arabs, and Isaac, who fathered Jacob (aka Israel, patriarch of the Israelites/Jews) and Esau (patriarch of the Edomites of the Negev).

Ham was cast as the father of the Egyptians and Canaanites, and Japheth as the father of the Greeks and Romans.