Quite a few countries do not allow dual citizenship. So a person who was born in the US and is therefore US citizen at birth will not be allowed to have that country's citizenship until they revoke the US one.
China and Singapore are some of the more prominent examples.
The Netherlands as well.
Both of your examples are wrong.
China considers it a "nationality conflict," the child is issued a Travel Document and treated as a citizen domestically, they can still be registered on hukou and get ID card. Apparently they used to unofficially force you to decide as an adult, but stopped a few years ago and now issue the Travel Document for life.
edit to add -- that assumes the parent is not a unconditional green card holder, which is the scenario here.
Singapore allows dual citizenship until 21. Which is not necessarily a good thing, as if you do not do their national service you will effectively get banned from ever going there even if you renounce it later.
Japan and Korea both allow it forever from birth in practice, but the latter also has some complexities regarding the military (either renounce before a certain age or you have some restrictions returning until past a certain age).