Corsica and Pays Basque are quite specific in that they still have some proponents for separation from France. Other areas are more about having local cultures while still being fine with belonging to a larger country. There aren't many people of Brittany that want to be independent, but most of them are proud to be Britton.
It's a bit earlier than that, it started with François 1st in 16th century. French was defined as the single official language of the country in 1539.
Also yeah the Gauls and the Franks were two different people (Franks come from the German area IIRC). France takes its name from the Franks because they were the first to unite the country with a single authority, while the Gauls were a mix of independent tribes.