I worked in a DC office years ago where I handled visa issues upon occasion. In handling a case involving a SKorean dual citizen, I was told by State department liaison that the US does not recognize dual citizenship, even if another country may. Should a person having a dual US passport get into trouble - in this case SKorea (ie. for not serving the mandated military service), the US would not intervene in any way to help.
As I understand it, only other some other countries recognize “dual citizenship” - the US does not. In the case of Hill, she probably did not have to renounce her foreign citizenship when she took American citizenship - but the US would ONLY recognize her as a US citizen at this point. If a US citizen takes an oath to another country, as far as the US government is concerned, they have renounced the US citizenship.
Some people insist they have “dual citizenship”, but it is only in another country’s eyes (and courts).