As far as I can tell, initially the term Jew strictly referred to a patrilineal tribal identity.
If by initially you mean before the kingdom split. And by Jew you meant Judean.
The 12 tribe community was known as Israelite.
13, but let's say 12.
One had to have patrilineal descent to become an Israelite. So for example, being the son of the Israelitish woman and Egyptian man didn't make one an Israelite or Danite.
This was only true before the Torah was given. After were received the Torah, our rule became that Israelite status was passed through the mother, but tribal identity was passed through the father. One could no longer become an Israelite through patrilineal descent as evidenced by Ezra 10:3. There it says that everyone who "fears the commandments of G-d" are told to send their non-Israelite wives
and children. Had we still followed patrilineal descent those children would be Israelites and not required to have been sent out.
After having received the covenant that the Torah, it also became possible to to come under the Mosaic covenant and become an Israelite through conversion as well. Then one would become a "ger" - one who converted to the Israelite religion, having the same status as any other Israelite.
Otherwise, please show me the earliest clear use of the word Jew / Jewish to describe someone from another tribe in the Bible.
The area of land under control by the Judean Kingdom was called "Yehudah (Judea)" (even though it included more than just the tribe of Judah's portion). The language they spoke was called "Judean". And the people were called Judeans. See Jeremiah 52:27 and onwards and Esther 2:5-6.
As far as I can tell, it was after the northern kingdom perished and the tribe of Judah was promised it will not (perish) that the term Jew/ Jewish began to refer to members of a nation (without regard to patrilineal descent).
Way back in 2 Kings 18:26, 28 the language spoken in the kingdom of Judea, Judean. They lived in a land called Judea. They spoke a language called Judean. And later we find the people being called Judeans. You see the trend?
Then gradually from there it began to refer to a religion to which one could convert. Prior to that one's tribal identity was one's religious identity.
The concept of a convert is already discussed in the Torah. There are two types of "ger" mentioned in the Torah. Colloquially we call them "ger tzedek" - the convert and "ger toshav" - the one who is permitted to live among the Israelites in Israel.
This is readily identifiable by two otherwise contradictory verses:
"For the congregation, there is one statute for you and for the
ger who dwells; a statute for generations. As it is for you, so shall it be for the
ger before G-d. One teaching (lit.Torah) and one judgment shall be for you and for the
ger with you."
- Num. 15:15, 16
"Do not eat any carrion. To the
ger that is in your gates give it to eat or sell it to a stranger. Because you are a holy nation to G-d."
- Deut. 14:21
As you can see the first verse says that the "ger" and the Israelite would have the same Law. But the second verse says that the Israelite and "ger" would have a different Law (ie. one can eat carrion and one can't). The first verse is talking about a convert: he is no different than a Jew. The second is talking about a non-Jew who is permitted to dwell in Israel (in your gates - as opposed to his own gates). This is as opposed to another verse that states:
"They shall not dwell in your land, lest they cause you to sin to Me. Because you will serve their gods and it will be a trap for you."
- Ex. 23:33
This verse provides the prohibition for Jews to allow non-Jews to remain in Israel along with the way to get around this prohibition: if they are not serving idols, but the Israelite G-d. Then they gain the status described in Deut. 14:21 called a ger toshav and they may remain in the land and we are even commanded to support them. The present incarnation is called Noahide. However they are still not considered Jewish as they are permitted to eat carrion - unlike the group mentioned in Num 15:16 who has to follow the same Torah Law as any other Jew, because they are converts.
Perhaps this is why you are also confusing the Quraysh as a member of the Ishmaelite nation with a tribe that has to have an Ishmaelite patrilineal descent.
Nope.