My experience growing up in the U.S. was that there is a strong desire to inculcate a certain Americanness there as well, and resistance to people failing to integrate and adopt it, though it might not be to the same extent. As a superficial but very visible aspect, we started each day in elementary school by saying a Pledge of Allegiance to the American flag. There was also a strong focus on American civics and history throughout schooling, often with a general tone of American exceptionalism, and a normative view of what it means to be an American. In high school in Texas, this was even supplemented by a normative view of what it means to be a Texan, though those of us not from Nth-generation Texan families tended not to take that part seriously by that age.
As far as schooling goes, Denmark is more liberal than the U.S. when it comes to choice, because there are school vouchers (not sure if true in Sweden). If you prefer not to use the state system, you can get a voucher equal to the prorated per-student cost of public education, to spend on private education instead. One of my colleagues uses this to send her daughter to an English-language school, rather than a Danish-language school. The U.S. doesn't have any similar program that will let you choose a Spanish-language school over an English-language school (and there is huge political pressure against Spanish-language schooling, because it is seen as a failure to integrate into proper Americanness).