It also included refugees from the Northern kingdom, so that when the Kingdom of Judah was carried off into captivity, it included all 12 tribes plus Levi. In other words, the word "Jew" can refer to all of the tribes.
You are not mistaken (in a sense) that all tribes of Israel are sometimes referred to as Jews.....but incorrectly as it is. They are never called by that appellation in scripture and indeed sometimes it's very obvious they shouldn't be.
King James [II Kings 16:5-6]
Then Rezin king of Syria and Pekah son of Remaliah king of Israel came up to Jerusalem to war: and they besieged Ahaz, but could not overcome
him. At that time Rezin king of Syria recovered Elath to Syria,
and drave the Jews from Elath: and the Syrians came to Elath, and dwelt there unto this day.
As you can see....scripture defines Israel apart from the Jews. In fact this is the first place in the Bible where the word, "Jew" is used.....and they are at war against "Israel". All Jews are Israelites (as all Californians are Americans)......but not all Israelites are Jews (as all Americans are not Californians).
In the Early Church these points were well understood by the Church Fathers but as time went on and tradition took over many things became obscured. I still hear frequently statements such as...."Oh yes...the tribe of Dan, who were one of the Jewish Tribes...................." No they weren't.
Justin Martyr said in his first apology:
CHAPTER XXXII --Moses then, who was the first of the prophets, spoke in these very words: "The sceptre shall not depart from Judah, nor a lawgiver from between his feet, until He come for whom it is reserved; and He shall be the desire of the nations, binding His foal to the vine, washing His robe in the blood of the grape." It is yours to make accurate inquiry, and ascertain up to whose time the Jews had a lawgiver and king of their own. Up to the time of Jesus Christ, who taught us, and interpreted the prophecies which were not yet understood, [they had a lawgiver] as was foretold by the holy and divine Spirit of prophecy through Moses, "that a ruler would not fail the Jews until He should come for whom the kingdom was reserved"
(for Judah was the forefather of the Jews, from whom also they have their name of Jews); and after He (i.e., Christ) appeared, you began to rule the Jews, and gained possession of all their territory. And the prophecy, "He shall be the expectation of the nations,"