I don’t think any other countries actually use the “natural born” qualification, it seems to be something unique to the United States. There were similar types of citizenship in the past, at least in Britain, but they were called “native subjects” or something like that, and it was based on the nationality of their father.
More importantly though, whether another country designates someone as a citizen, of any variety, has absolutely no impact on their citizenship status in the United States. We are a sovereign country, and citizenship here is based on our laws alone, without being influenced by the laws of other countries.
For example, I have a friend whose grandmother was from Ireland, and due to Irish law, he can claim Irish citizenship. That doesn’t mean he loses his American citizenship, just because Ireland passed a law which considers him a citizen of their country. Our laws determine his citizenship status here, and the Irish laws are irrelevant to the question because they carry no weight in our territory.
If you study immigration cases, you'll find that what you say is not true. In mixed-nationality cases, it is not unusual to look at the laws of other countries, to settle the question of citizenship to the person. Of course, that is done from a basic framework of US law, but just to pick an example, the US residency requirements for a one-citizen parent vary, depending on if that citizen parent is in wedlock or not, as recognized in the foreign country. There is a pretty long string of cases "decided the wrong way" because the government misled the court about the contents of Mexico's constitution, when the controlling law (of matrimony or child-care) was actually that of a Mexican state. The outcomes of those cases are citizen vs. not-citizen.