Mordecai was considered a Jew, because he was from Judah. He was one of the captives taken when Judah fell to the Babylonians.
The kingdom of Judah consisted of the two southern tribes, Judah and Benjamin, either of which would have been called Jews because of their nationality.
We are making progress. You consent that the term Jew does not only refer to a Jew of the tribe of Judah, but whoever attaches himself to the nation of the Jews, ie., the Jewish People. All Israel, the remnant of Israel from each of the Twelve Tribes, in Exile or the Land of Israel, looked to Judea because therein lay Jerusalem, the eternal city where God placed His Name.
Mordechai was of the tribe of Benjamin, not the tribe of Judah. He was a Jew. There were many who saw the salvation of the Jews in the time of Mordechai and Esther and became Jews while in the Exile. And in every province, and in every city, whithersoever the king's commandment and his decree came, the Jews had joy and gladness, a feast and a good day. And many of the people of the land became Jews; for the fear of the Jews fell upon them.
Esther, Catholic chapter eight, Protestant verse seventeen,
as authorized, but not authored, by King James,
boldness mine