There's another error (at least one I noticed); they don't mention
Collectivité territoriale de Saint-Pierre-et-Miquelon, French islands off the coast of Newfoundland, Canada. How do they fit in this complexity?
And Ulf, GGR and others--a good deal of the peace over the past 70 years has to do first with the collective "enemy" of the cold-war, the Russian-Bloc nations behind the once "Iron-Curtain" which of course engendered United States military troop commands numbering in the hundreds of thousands, not to mention the armament, air forces and related. Kind of hard to start a war when this kind of enforcement is around!
But, all of Europe has not seen 70 years of peace. Don't forget the war in Kosovo; "armed insurgency" in Macedonia; the Russion invasion of Hungary in 1956; of Czechoslovakia in 1968. Bosnian war, Croatian War of Independence, Romanian revolution…while not world wars, or even wars engaging all of Europe, they did make travel dangerous in certain areas, and resulted in loss of life. We here in the USA don't always get all the stories, but one we will certainly hear and see again is the remnants of the destruction of Sarajevo--which hosted the 1984 Winter Olympics (that's why we'll hear about it) but was torn apart with the breakup of Yugoslavia…not precisely 70 years of peace.
The complexity of the EU is almost matched by two things: first, the United States Forces in Europe. Look THAT up and you'll find a variety of interesting acronyms such as USAREUR, AFCENT, USEUCOM, SHAPE, AFNORTH, AFSOUTH, and more--an ever-changing alphabet soup typical of a changing military command structure starting with the end of WWII, incorporating NATO and the UN, too.
Second is our own United States; the EU has a small number of countries, we have 50 states plus the District of Columbia, some overseas territories (such as Puerto Rico, Samoa, Guam, Virgin Islands, etc.) and the same kind of humorous complexity offered by the EU video could be made about our own USA! Maybe it has and we just have not found it yet.