Which exile are you talking about?

This is a really interesting question, not least because of the enormous amount of historiographical questions it raises. To answer all of them would be impossible, but I hope I can give at least a summary overview of the biggest ones.

The foremost one to tackle is the purely historical question of the development of Israel and Judah until the destruction of Jerusalem in 587/6 BCE. Both nations developed relatively independently, though mostly as vassals of larger empires, and Judah possibly paid tribute to Israel before the Assyrian invasion of 712ish. Going back to the 10th century BCE or so, however, there are no surviving contemporary historical records about them at all. Archaeological evidence suggests that a semi-nomadic, pastoral ('tribal') system lies at the basis of both nations, as it did for most Levantine states, but whether this system resembled in any way the 'tribal conference' described in the Hebrew Bible (esp. Joshua, Judges and 1 Samuel) is anyone's guess. This relates to your question in two important ways: 1) because there was no formal unification of Israel and Judah before the Persian period, and because no contemporary records remain, the notion of a consciousness of shared tribal history in pre-exilic Israel or Judah is impossible to prove; 2) 'Hebrews' aren't really a thing until the Judahite authors of the Bible come up with them - and they call the language of Judah 'Yehudit' rather than 'Ivrit'.

This leads rather nicely to the second important question, and that is what happened during the Babylonian Exile, and to whom. The Exile itself was a banishment of elite members of Judahite society to Babylon (NB: Israel had ceased to exist as a state after the Assyrian destruction of Samaria and never recovered its being reduced to a province). The composition of what would become the Hebrew Bible, therefore, was largely a southern affair that rewrote history to be focused on Jerusalem, and any understanding of tribal history must be seen through that lense. The appropriation of Israelite history (again, to whatever extent that matches reality we don't know) is part of this process, and this makes it significantly more difficult for us to know what might have been part of a Judahite's or Israelite's awareness of pre-monarchic history.

Post-exile, the scriptures that would later be canonised as the Hebrew Bible become the paradigmatic understanding of history for the people who start calling themselves Hebrews; 'Israel' - and with it, the idea of a common tribal history - is codified into this new tradition as a fundamental part of their national identity. If we define a Hebrew as someone who ascribes to this ideology, they would most likely be aware of this idea, even though fundamentally, we don't know to what extent this reflected actual historical fact.

Sources:

Dever, W. 2001. What did the biblical authors know and when did they know it?

Thompson, T. 1992. Early History of the Israelite People.

Finkelstein, I. 1988. The Archaeology of the Israelite Settlement.

Accessible source: Finkelstein, I. and Silberman, N. 2001. The Bible Unearthed.