As pointed out above, that isn’t true for everyone. The U.S. doesn’t like its citizens from having dual citizenship but it can’t stop it either. There are plenty of people that have dual U.S. and (other country) citizenships. That doesn’t mean that it is risk free however and the U.S. doesn’t want its citizens to collect citizenships on a whim like trading cards. If there was ever a real conflict between the two countries that involved something like a draft, it could put everyone in a bind. It could also create a problem if the person ever got into serious problems with a third country while traveling. In short, the U.S. won’t just strip you of your citizenship if another country offers you another one but it isn’t too hot on the idea.