Picard is from 'France' as is stated in the show....however that does not mean 'France' in the modern sense. It is simply a region, a geographical area. NOT an independent, autonomous body.
Problem with that is, once France ceased being a political unit/country, how much longer would little Le Barre continue to be "
We seen in the past few decades where a country has ceased to exist, broken up, that the people fairly quickly stop identifying with the former large unit.
Just because a governing body is established at the world wide level, doesn't mean the countries that assembled it discontinue to be sovereign countries.
It all has to do with how much (and how little) United Earth is allowed to do. If UE's main duty is to speak for Earth on interstellar matters, and has few other abilities and responsibilities, it could be a fairly restricted administrative body.
Nothing says that UE actually can govern the entire surface of the Earth.