U.S. Law requires the signatures of both parents on a passport application. If one parent is absent from the relationship, then it is impossible or extremely difficult for a minor to obtain a U.S. passport.
If one of the parents is not a U.S. citizen, then the child may become a dual citizen or national of the country of the parent with foreign nationality and a U.S. Citizen. Consequently, the parent may be able to obtain a non-U.S. citizen passport.
U.S. Law requires U.S. Citizens to exit and return to the U.S. with a U.S. passport. But, there are no exit controls in the U.S. as they pertain to the U.S. State Department. Consequently, minors with dual citizenship are allowed to travel internationally on a foreign passport. When the minor returns, immigration will notify the State Department a child with foreign nationality has returned to the U.S. without an U.S. entry visa. A record of the child’s entry into the U.S. with a foreign passport and no entry visa will be created and stored.
The two parent signature on a child’s pp is not correct. I got my first pp at age 10 - my parents were divorced, and my father lived in a foreign country and didn’t even know that I was getting a pp.