Europe beyond the Continent of Europe / Maybe closer to home?

New York City, because it's by far the closest thing to London. See the same shows on the West End or Broadway. Tap into the busy subway or Underground system, emerge at Wall Street or the City to see the international bankers swarming into their skyscrapers. Walk across that marvel of 19th century iron work, the Brooklyn Bridge or Tower Bridge. See genuine medieval architecture at the Cloisters in Upper Manhattan, or various London sites. Have coffee at Pret a Manger. See treasures of ancient civilizations and European art at the Met, or at the British Museum and National Gallery. Visit virtually identical museums of natural history. Feel the multicultural energy of a city of many immigrants and enjoy their diverse restaurants. Take a sightseeing boat on the Thames or the Hudson. Most people speak English but their accent may be a bit different than what you're used to. Hail one of the ubiquitous black (or yellow) cabs. Beware of proliferating bike lanes with speeding scofflaws endangering pedestrians. Have sticker shock over the high price and small size of your hotel room. Take a nice walk in Central or Hyde Park. When in Central Park, take a look at the Egyptian obelisk behind the Met -- very similar to the one on the Thames Embankment. Feel the busy pulse (but tawdriness) of Soho or Times Square. You can take a train or bus east to Greenwich, but the London one has a lot more to see than the Connecticut one, frankly. If a pint and a pie in a pub is what you crave, London will deliver that, but so will NYC (a very good place on E 76th St, for example). Fancy a large aristocratic mansion housing a magnate's well curated art collection? Try, e.g., the Frick or Morgan Library in NYC, or the Wallace Collection in London. Interested in a World War II naval ship? London has the Belfast and NYC the Intrepid.

You've hit on something very interesting. There are borderlines that are difficult to distinguish.
You've mentioned Guyane, a department that had a well-known politician in the Holland government, a department that attracts illegal crossings from its neighbors in Brazil and Suriname, seeking healthcare, jobs, housing, etc.

One day on Gudaloupe, a French island department in the Caribbean, I saw a sign in a national forest saying an EU restoration project was going on. "Your EU funds at work" but in French.

At this moment the residents of Mayotte, an Indian Ocean French territory ravaged by a cyclone, are the leading story in the French news every night, fifteen minutes of information. If you listen to the people interviewed, they are speaking French. They went to French schools. They have the same governmental structures and aid as here in the metropolitan France.

These DOMs and TOMs (overseas departments and territories) have the same government structures and benefits as any department in the "metropole" as the French hexagone is called. They have blended, rich cultures with multiple identities.

One evening as Air France planes were landing en mass bringing French Metropole vacationers to Guadaloupe, I asked a local school teacher, a Guadaloupian, if he felt French, Caribbean, or Guadaloupian. With the Parisians deplaning, he said Caribbean at that moment. But he and others were preparing to go out on strike against their employer, the French national education system. In my opinion, that's a very French way to handle a labor dispute. There's a lot that makes up a nationality in Europe and in its overseas departments and territories.

A very interesting question. I know on Guadeloupe and Martinique I felt I was very much in France, but also in the Caribbean. But then I was on a small van tour direct from the freight port on a bespoke tour from the ship. So my experience may be skewed as opposed to if I had flown in or arrived by cruise.
In Ponta Delgada in the Azores I was unmistakeably in Portugal- but can't speak for outside the Capital.
On Sint Maarten (the Dutch side) I felt I could have been anywhere, but Saint Martin (the French side) I was totally in France. The difference in all respects was marked.
Curacao depended on where you went- in the city centre I felt quite distinctly in the Netherlands (much aided by being on a local tour I picked up by chance at the pier), but in the malls closer to the Cruise Port was anywhere, rather Caribbean, which may be kind.
French Polynesia I think depends on your island.
Nuku Hiva I couldn't make my mind up- in the Cathedral and the Bank I felt French, but elsewhere on the island very Caribbean.
Bora Bora just felt like Paradise- I could barely detect any French influence, other than in the Church.
Tahiti on the other hand felt very French on the surface, with lots of European funding signs. And French is the official language. I remember writing in my diary 'Paradise Lost' about Tahiti, to me not very South Pacific, very visible social issues. But other's mileage probably varies a lot.
Yet the public bus system on Tahiti was very Caribbean in it's lack of apparent organisation. Impenetrability of timetables and the way the timetables fell to pieces when it started to rain.

I thought we could have fun with this. But Slate blew my mind. YES. So obviusly yes, but I still would have never thought it.

It’s education, language or languages, and mentality, very difficult to discern how deeply the elements are or aren’t embedded in so many different people. But these places are all there and all have strong elements of the European home country. However, many of the European countries were created by uniting different populations that had their own languages and customs. As recently as WWI, some soldiers spoke only their regional languages and met people from other regions of their countries for the first time when called to fight for some reason they knew nothing about.

France has the 2 islands, Saint Pierre and Miquelon, off Newfoundland, Canada. Their small population is represented in France by 1 Senator and 1 member of the National Assembly.

Canada is great, we love Quebec City and Montreal. Also, love Ottawa.
Out west, the Canadian Rockies are fantastic, do Banff, Lake Louise and Jasper. Of course, do Vancouver and surrounding areas.

South America is great. It is cheaper to visit than Europe. Each country is different with different mini-culture and cuisine.
Do a cruise around Cape Horn from Buenos Aires to Chile for two weeks, Amazing.
Spend some days in Buenos Aires and Santiago, Chile.
Enjoy the best steaks ever in Argentina and fantastic seafood in Chile. Love the Chilean Sea Bass and crab and cheese filled empanadas.
Peru is my favorite country in S. America, do Lima, Cuzco, the Sacred Valley and Machu Picchu. Suggest taking a tour. We used PeruAgency.com
Do a Caribbean cruise and visit some wonderful islands like St. Thomas, Barbados, Antigua and Aruba.

I forgot to mention doing Alaska.
Take a cruise from Vancouver to Seward, Alaska, then do Denali.