Ettore Bugatti was indeed an Italian, but the car company he founded was not. In 1909, manufacturing of the Bugatti-branded motorcars began in what used to be a dye factory in Molsheim, near Strasbourg, which at the time was in Germany.
During the war, Ettore Bugatti was in Paris, designing aero engines for the Allies.
After WW1 Alsace returned to France, and with it, the Molsheim factory. The 1920s were Bugatti's heyday, and it was then the World's most exclusive and prestigious luxury brand.
At the time and till it ended its acrivities somehere in the grim 1950s (save for the WW2 hiatus, when it was taken over by the Nazis as the rest of the continent), Bugatti was a company based in France, manufacturing in France using French labour, and so was a French company making French cars.
After WW2 the attitude of the French government was favouring small popular cars (Renault, Citroën, Peugeot, Simca) and hostile to “bourgeois” motors, and so not only did the Delahaye, Delage, Talbot-Lago and other high-luxury makes disappear but the mid-range ones as well such as Panhard and Hotchkiss.
The attempt to found a new high-class French car marque, the Facel-Vega, failed as well alas.
In the 1990s some businessmen bought the rights to the Bugatti name and planned to make new cars bearing the proud name in Italy. Off course this is not a rebirth of the historical Bugatti.
Later, before any commercially-available car materialised, the outfit was acquired by VAG, which decided to get closer to the source of the old Bugatti cars, so Bugatti-branded cars are now once again made in Molsheim.
My favourite French manufacturer is Hotchkiss, which is not on the list.
That company itself was founded in France by an American, decades before automobile, making cannons. And so, Hotchkiss too was a French car manufacturer. Its last motorcar was the miltary Willys/Kaiser Jeep made under licence for the French army, which is quite a fitting end for what started as a war weapons company.