People in Saint Barthélémy are just regular French and EU citizens who live far away from the mainland. The territory does have some degree of autonomy (more than other overseas départements) but still less so than a US state relative to the US federal government.
In that, most French territories would have a status maybe more comparable to a county within a US state (afaiu the US system) than to a US state within the US. The whole comparison is a bit sketchy but France isn't a federal state and only New Caledonia and Polynesia have local legislation, all the rest of the country is very unitary in political and administrative aspects.
For instances, when there was all the talk about Barack Obama's birth certificate, it wouldn't have disqualified him as a candidate to claim his birthplace was in Puerto Rico. (See also John McCain, who was born in the Panama Canal Zone, which had a similar status to Puerty Rico, but distinct from American Samoa.) It would have disqualified his candidacy if they were able to prove he was born in Kenya or American Samoa, which would have made him not a natural born citizen, even though his mother is a US citizen, which would have Barack Obama a US citizen at birth, regardless of where he was born.
St Bart's, like New Caledonia and Polynesia, is an EU Overseas Country/Territory, and a French Overseas Collectivity. St Bart's isn't as close politically to France as French Guiana or Martinique, which are both French Overseas Regions, and EU Outermost Regions.
So I would compare places like French Guiana or Martinique to places like Puerto Rico, Guam, and the US Virgin Islands, and places like St Bart's, New Caledonia, and French Polynesia to places like American Samoa.
https://en.wikipedia.org/wiki/Special_member_state_territori...
Hawaii is more like Malta in EU, a full member state.