Correct sir, except they are not of the tribe of Judah. So technically not Jews. They do have common ancestors but so do Arabs too. The descendants of Abram/Abraham. One by a concubine who was chased off, and one by his wife. His offspring with his Wife led to Jacob, I can’t remember who the next gen of his other “wife” was. He was also blessed, but less so.
Yes, the 10 exiled tribes were not of Judah, but their descendants practice Judaism, which now means they’re Jews.
‘Jews’ is a nickname. In Esther Chapter 2 for example, Mordecai the ‘Jew’ is a Benjaminite. Judah (Judea) was the name of the southern kingdom, known to include remnants of other tribes, as well as more than its share of the tribe of Levi, which included Kohanim.
Technically, the proper term is B’nai Yisrael or B’nai Ya’akov.