Franks identified themselves...well, as Franks or Franci, as their main collective identity.
This term is present everywhere in contemporary texts and unmistakably designs the inhabitants of the "Frankish core", so to speak, made of the sub-kingdoms of Neustria, Burgondia and Austrasia : not that, critically with the VIIIth century, there weren't regional identities stemming from the association of local aristocracies to sub-royal palaces (the Chronicle of Fredegar being rather Austrasian in its support, the Book of the History of Franks rather Neustrian and loyalist) but never to the point challenging a common Frankish identity in any late Merovingian or early Carolingian source.

Peripheral ensembles such as in Aquitaine (where people considered themselves as Romans), Thurgingia, Bavaria, etc. are a mixed bag : they did had populations creating or re-creating regional identities distinct from Franks, but not to the point they necessarily thought themselves entirely outside the Merovingian world, especially as their own elites were familialy, politically and culturally close to "core" Franks. Eudon, who managed to rule Aquitaine as princeps (basically claiming the same kind of power there than a Merovingian king) had a Frankish name and at least acknowledged some Merovingian and early Carolingian over-lordship (compelled and forced in the last case, arguably).

Early Carolingians replaced most of local aristocracy with their clients or family, increasing the Frankish character of Aquitaine, Provence, Alemania, Thuringia and Bavaria (arguably being less prone to do so in Saxony and Italy, to attempt reaching a compromise with locals).
When Charlemagne plan in 806 the split of his empire between a Francia auditioned of most of Germania, an Aquitaine additioned from Burgundy, and an Italy additioned from Bavaria; he doesn't consider giving "ethnic" kingdoms to his sons, but "territorial" kingdoms, that would remain Frankish while dominating distinct peoples and gives them authority over Frankish nobles and territories as well; just as he's King of Franks and Lombards.
When Peppin II revolts against Louis I's decision to confiscate his kingship and gives it to Charles, he doesn't lead a national revolt, but a Frankish nobiliar revolt over a territorial (and particularist, truth to be told) sub-kingdom.

The situation arguably changed with the split of Carolingia into de facto distinct kingdoms (whereas Merovingian splits didn't implied the utter division of the regnum), but as every Carolingian king was Frankish, in a first time nobody really knew how to name the inhabitants of their kingdoms : Charles was reputed ruling over the "Carlensenses", the people of Charles' kingdom; and of course the "Lotharingia" of Lothar II.
During the IXth century, however, territorial and ethnic identities appeared in royal titulature : several western Frankish kings were named "King of Franks and Aquitains" up to the XIth century, Charles the Simple was acknowledged in 898 as "King of Franks, and Aquitains and Goths", (meaning the region between Rhone and Ebre), and Raoul "King of Franks, Burgundians and Aquitains".
Late Carolingian official titles being generally non-descript, "king" or "august emperor", previously mentioned titles are more given than assumed, but rex francorum dominates almost systematically when territories and "nations" are mentioned)
It is less an evidence of distinct and competing identities, than competition over territories becoming more and more independent (and where the temptation of a separate regnum as in Provence, is quite present) and an affirmation of royal legitimacy over them (either due to sheer claim, either due to a local support base)

While the Frankish identity lost ground in Germania (except in Franconia), it is less an ethnic revival, than a new political aspect of the situation as "national" dynasties emerge. On the other hand, western Francia kept a whole Frankish identity over regional and political ones (to quote Gavaudan, "Angevins, Bretons, Cambresis, Provencals, Poitevins" etc.) that would evolve trough the significant political, social and cultural change of the high middle-ages and became "French".