"Now that I know so many of the legacy brands are French, the fact that French’s is called “French’s” and describes itself as “American flavor in a bottle” is kind of funny (and American)."

According to McCormick's (owner of French's Mustard brand) website:

"George and Francis French, sons of R.T. French introduced French’s Classic Yellow Mustard, served over hot dogs at the St. Louis World’s Fair."

https://www.mccormick.com/frenchs/story

So, named for the creators in 1904, not for a specific association with the country.