They most certainly do and are called Franks both historically and today. Gallic notions were post-nationalistic understandings, they are political in nature and in some ways, a cope for the failure of France to truly fulfill its destiny as Lords of Europe as Pierre Dubois once spoke of in 1298. In other words, it is an acceptance of an insular relation to its other historic partners, namely Germany, Italy, Middle Francia, etc...
The Frankish identity was inherently a universal rendering by the Middle Ages surely. Even as a byword in other lands, it referred to a united Europe, such in the Arab world or some of the sensational Papal praising of the Empire and Crown of France.
Latin too, is more or less co-equal to Frankish at least on the continent. Through the medieval era, the Papacy never once distinguished between the crown that Clovis I wore in 509 and that of the kings of Paris. Nor did they understand the French tongue as a new tongue different from Frankish (same goes for the forms of German then common). This seems odd, but it is due to most western audiences being unfamiliar with the idea of true cultural intermingling and fusion. What occurred in post Western Imperial France and Germany, was a German to Latin version of what occurred in ancient Mesopotamia between Akkadian, Sumerian and lower Hurrian. They absorbed together and became indistinguishable to the folk at the time.
The difference, the break of the Papal universal rule over Europe, the decline of universal entities within Europe (the Empire, Angevin Realms and traditional French/West Francian kingdom) and its repudiation, led to new cultural movements, that instead of affirming older notions of cultural fusion and universality, embraced cultural and societal localism and a thin veneer of commonality. Such cultural assertions never existed in ancient iraq, hence if one looks to the records, one finds legitimately no difference or point to break Akkadian from Sumerian or vice versa except in strict and meaningless linguistic categories.
@funnyhat this goes for you as well regarding your recent posts on the topic.