To be clear, not as a term the Hittites gave the Greeks, but as a Hittite effort to transliterate what the Greeks - or at least some Greeks - were already calling themselves. Likewise, Homer uses Danaoi which is probably reflected in Bronze Age Egyptian 'Denyen.'
I've seen postulations that Achaean might also pop up in the Egyptian writings about the Sea People as Ekwesh.
The nature of the Linear B (=Bronze Age Greek) corpus is such that you don't get 'slam dunk' evidence either way but broadly I'd suggest it is plausible and perhaps likely that Greek-speakers recognized themselves as an ethnic group fairly early. It just wasn't the most important identity.
Among other indicators, while we're here, is that the bronze age Greek palace cultures have similar material culture - indeed, it is common to point to the emergence of more distinctive regional material culture after LBAC as a sign of the breakdown of interconnectedness.
On balance, this strikes me as most likely someone over- and perhaps misreading someone they heard attempting to stress that the Greeks tended to consider themselves Athenian/Theban/etc first and 'Greek,' like, fifth and that panhellenic thinking intensifies notably post-480 and taking it too far.
But the very oldest piece of surviving Greek literature is the Iliad and it already demonstrates a sense that the Greeks (Achaioi, Danaoi, Argeioi) are a distinct, identifiable group who can be distinguished clearly from other (Trojans, Ethiopians, Thracians, Amazons) ethnic groups.
When did the Greeks first develop the idea of "barbarians" for those who spoke something else?
Is it comparable to how, say, “nationalism” didn’t quite exist on medieval Europe, but at the same time, observers still noted clear distinctions between broad groups of people (like Germans vs. Italians) even when each individual group was not fully unified either politically nor linguistically?
Came here to ask that.
I was actually under the impression that the Greeks got their name from Helen of Troy (as the Trojan War was the first time the Greeks were semi-unified under the Amphictyonic League), but apparently I was way off on that.