In the Septuagint, "Kingdoms" 3 & 4 (= I Kings and II Kings), the various Israelite kings are called kings of Israel or kings of Judah (Iouda), not "kings of the Jews."
That was after the kingdom split under Rehoboam, the son of Solomon. The northern ten tribes retained the name “Israel” under Jeroboam’s rule, while Rehoboam’s southern kingdom was named “Judah” after the tribe the Jews came from, which also included the tribes of Benjamin and Levi; Jeroboam founded his own priesthood from “the lowest of the people” (1 Kings 12:31 and 13:33), who quickly turned to paganism.
Judah of course was the name that the Romans rendered “Judaea”.