Frankfurt was made site of the elections in the fourteenth century when Francs were chiefly the people in France and the Saxons were chiefly the people in Saxony (the one around Dresden). It's clear that when Martin Luther in his Tischgespräche speaks of Saxon, he does not mean a coastal Low German dialect but the language of the Prince-Elector who provided him refuge (as their language could be understood "both in the Nether- and the Upperlands" --- which is why, purportedly, he chose it for his translation of the Bible.) Could there be French-Frankfurters and Saxon-Frankfurters? Of course. But the people in Frankfurt itself had no reason to identify with any "Franks".

In earlier times, back when the empire was called Frankish, there appears to have been a notion of an earlier Frankish identity. Monk of St.Gall (9th century) does describe the particular "Old Frankish" dress:

»The mention of the trailing garment that the emperor wore at night has diverted us from his military array. Now the dress and equipment of the old Franks was as follows:—Their boots were gilt on the outside and decorated with laces three cubits long. The thongs round the legs were red, and under them they wore upon their legs and thighs linen of the same colour, artistically embroidered. The laces stretched above these linen garments and above the crossed thongs, sometimes under them and sometimes over them, now in front of the leg and now behind. Then came a rich linen shirt and then a buckled sword-belt. The great sword was surrounded first with a sheath, then with a covering of leather, and lastly with a linen wrap hardened with shining wax.

The last part of their dress was a white or blue cloak in the shape of a double square; so that when it was placed upon the shoulders it touched the feet in front and behind, but at the side hardly came down to the knees. In the right hand was carried a stick of apple-wood, with regular knots, strong and terrible; a handle of gold or silver decorated with figures was fastened to it. I myself am lazy and slower than a tortoise, and so never got into Frankland; but I saw the King of the Franks in the monastery of Saint Gall, glittering in the dress that I have described.

But the habits of man change; and when the Franks, in their wars with the Gauls, saw the latter proudly wearing little striped cloaks, they dropped their national customs and began to imitate the Gauls. At first the strictest of emperors did not forbid the new habit, because it seemed more suitable for war: but, when he found that the Frisians were abusing his permission, and were selling these little cloaks at the same price as the old large ones, he gave orders that no one should buy from them, at the usual price, anything but the old cloaks, broad, wide and long: and he added: “What is the good of those little napkins? I cannot cover myself with them in bed and when I am on horseback I cannot shield myself with them against wind and rain.”

...

Now, when I said Frankland just above, I meant all the provinces north of the Alps;«

Some 50 years earlier Eginhard did not have a notion of the "Old Franks" yet, but he did describe quiet well that the native language and traditions of Charlemagne himself were in decline and different from those of "the Franks":

»He also wrote out the barbarous and ancient songs, in which the acts of the kings and their wars were sung, and committed them to memory. He also began a grammar of his native language.

He gave the months names in his own tongue, for before his time they were called by the Franks partly by Latin and partly by barbarous names. He also gave names to the twelve winds, whereas before not more than four, and perhaps not so many, had names of their own.«