As /u/Mycd mentions, blue was a major part of the identity of the French monarchy. However, it wasn't directly tied to the concept of France itself. During the Hundred Years War, France was in a dire position, slowly it was being chipped away and alliances were shifting away from the French King to the English. As a result, the need of a strong French identity was needed, which involved two major images, blue and the fleur-de-lis.
Both of these are a major part of the images and paintings of Jean Fouquet, an artist that was employed by Charles VII and Louis XI. While blue isn't a solely French color, it's frequently employed in the art of Fouquet, particularly in respect to images of French Kings. With Fouquet, the identity of France is on display, particularly in the Grandes Chroniques de France which was commissioned by an unknown patron.
In this illuminated page, Philip II enters Tours with the blue field and gold fleur-de-lis being far more present than any other color on the page. This is continued several times in the Chronicle.
However, Fouquet would continue to work as a court Painter during the reign of Louis XI. His placement would make him one of the main individuals whom would help in development in French identity through art. All of this discussed by Erik Inglis in Jean Fouquet and the Invention of France.
Around 1144 the Basilica of St Denis was the first real usage of blue colored stained glass (cobalt) and was known throughout Christiandom as "bleu de Saint-Denis", and later installed in other prominent churches like Chartres Cathedral and Sainte-Chapelle. The color blue evolved to represent the Virgin Mary in these centuries as well, symbolizing holiness and virtue, the sky/heaven, etc. As a cloth dye, it was a rare and expensive import from Asia, from ground-up Lapis lazuli ore, so only the wealthiest had blue garments and coats of arm.
I think it was King Louis IV (1214-1270) who really popularized blue though, was his prefered color for dress, and is when the coat of arms of the kings of France became an azure or light blue shield with fleur-de-lis. (here's his funeral tent https://en.wikipedia.org/wiki/Louis_IX_of_France#/media/File:Mort_de_Louis_IX_le_Saint.jpg for example)
Everyone after, and even in other parts of Europe, blue quickly became a royal color, implying dignity and wealth. In Italy for example:
dyeing of blue was assigned to a specific guild, the tintori di guado, and could not be done by anyone else without severe penalty
Back to France though, it was essentially popularized by Louis, with all the greater houses and families incorporating it into their seals and coats of arms. https://en.wikipedia.org/wiki/Gallery_of_French_coats_of_arms
TLDR : Blue was for a time a very rare and expensive and of religious interest, and one of the last and hardest colors to be made into dye, so elites and royalty adopted it as their colors and wore as clothing prominently.
