Is it accurate to call Abraham a Jew? It is common to associate Abraham as "father of the faith" and the first Jew.

However, since  Abram's family came from the land of Ur, isn't it more accurate to say that he wasn't a Jew but - in reality - a Babylonian that became a Jew because of his faith in God?

This is open for answer by all denominations. I would like to hear their thoughts on the matter.
The word “Jew” follows etymologically from the name “Judah”, Abraham’s great-great-grandson. Judah is Jacob’s fourth son and the fourth son of Jacob’s fist and less favored wife Leah. The second king of Biblical Israel is his descendent.
The first king was Saul, a descendent of Judah’s youngest brother Benjamin. But Saul sinned against Yahweh and while Saul was still king, he anointed David of Judah to be King of Israel by his prophet Samuel.
After David’s death, the kingdom of Israel split into two Kingdoms upon the succession of his grandson Rehoboam, the son of David’s son Solomon who built Yahweh’s temple in Jerusalem, though his Temple really resided in Solomon. The northern Kingdom retained the original name, Israel, and the Southern Kingdom was known as Judah. The Jews are judahites in that sense- cultural and ethnic successors of the southern kingdom. Both kingdoms were eventually lost to conquering empires.
Abraham was among the first people to practice what we would call the Jewish faith in earnest- monotheistic veneration of Yahweh, who he knew as El Shaddai, “Almighty God”, though not all of Abraham’s offspring are counted among his children according to the promise Yahweh made to him. Through much of the Biblical Narrative, Yahweh is called “The God of Abraham, Isaac, and Jacob.” It is not accurate to say he was the first Jew, since the name comes from his own offspring. It would be more appropriate to refer to him as the father of the Jews.