No, it is based on "nation", having a shared culture, language, and, yes, values too, but it's not just values. An Italian or German or Englishman can be just as white as a Frenchman, but the average Frenchman will not consider them French.
Part of it is pride in the accomplishments of your nation, and attachment to the fellow members of your nation. A Muslim must consider himself part of the Islamic Ummah more than he could consider himself a fellow to Christian French.
Yet there are several million Frenchmen of British, Spanish, Italian, German, Flemish, Polish, Corsican, and Portguese ancestry.
And they are considered Frenchmen. Very few Frenchmen would say "Chopin? MacMahon? They weren't real Frenchmen."
The borders, language, and culture of "La France" took on their classic acceptation only fairly recently.
France is a deliberate cultural construct, not a Platonic Idea come to life.
Yet he grew up in Algeria, the son of a Spanish mother and an Alsatian father (who was born in France, but grew up in a German-speaking home) and did not set foot in Europe until he was an adult.
Also the father of French New Wave cinema, Jean-Luc Godard, grew up in a Swiss Protestant household and was of Swiss-French, Swiss-German, and Danish background.
They both adopted - and helped create - mainstream French culture - and they are considered purely French.
The average Frenchman does not view them as interlopers, or somehow "off."