US does allow dual American citizenship if the other country also allows it. But technically, a US citizen's citizenship in another country is not recognized. It's kind of funny because the government knows it, but just ignores it.
The only exception is if a US citizen wants to work in a government agency where intelligence and highly sensitive information will be shared. Then, naturally, it is required for the US citizen to drop the other citizenship. The FBI won't even process someone's application for a job that has access to intelligence, if the applicant doesn't drop the other citizenship first.