The Bible is very precise in its wording.
The Bible says that Jesus is a Galilean, a Nazarene from Nazareth and of the tribe of Judah, a Judahite Israelite. His lineage goes back to Judah. He is a Hebrew.
The term Jew, invented in 1775, is not equivalent to the word Israelite. In the old testament it was applied only to the two Southern tribes of the house of Judah. Mostly in the NT it should have been translated Judean (as in country, not race or people). It is the same as you would use Texan to mean anyone living in Texas. Jesus could be called a Judean because he was born there. As an adult He never lived in Judea. Very few Israelites actually lived in Judea at the time of Christ. Most Israelites lived to the North in the old territory of the ten Northern tribes. The woman in Samaria called Christ a Jew, but that does not mean she was correct.
Today, the term Jew is applied to an unrelated people as they admit in historical and official writings.
“Strictly speaking, it is incorrect to call an ancient Israelite a “Jew” or to call a contemporary Jew an “Israelite” or a “Hebrew.” The first Hebrews may not have been Jews at all,“ The Jewish Almanac (1980)