Translation:The United States has the world's largest stock exchange.
Both options should be available. UK-en (perhaps also AU, CA as well) treats collective entities as plural nouns (mostly?), while US-en treats collective entities as singular nouns (though not always adhered to colloquially, particularly with respect to bands). As a US-en speaker, I'd say things like, "Microsoft is promoting its new product," or, "The United States is never going to be the same." Formally, anyway.
Just because the most common phrasing has shifted from plural to singular since the 19th century doesn't necessarily mean that the plural form is wrong now.
If this was an English course, I could understand why only the most commonly used version should be correct. But in a Norwegian course, both should be accepted, even if one of them is old-fashioned.
Good point on possessive. I'd guess it's because an apostrophe-S would appear to change a plural noun no singular, so it'd extend to similar things like The United Nations. But The League of Extraordinary Gentlemen could add an apostrophe-S with no issue. (That also has "League" earlier on as a main now, but thinking up some sample sentences, I don't think it affects anything here.)
This BBC article goes on to say that UK-en makes a distinction between the country as a country = singular, and the government of the country = plural, so apparently even those speakers may occasionally use The United States as a singular.
Yes in French "Bourse" means both "børs" and also "coin purse" (it refers to old coin purse when it was a little tissue full of gold, like in Robin Hood). We still use "bourse" to refer if we have money or not : I don't have nothing anymore in my "bourse" ; his "bourse" is full (even if it's a bank account).
I must admit something strange... I actually learned today, with this sentence the english word "Stock exchange". Probably I heard it many time , without understanding the meaning. Of course I know Wall Street. But not "stock exchange". And I really don't know how I could escape this english word 40 years...