No? If you're born in the US you have US citizenship, you're American. You don't just magically get citizenship for your parents home country, at least not for most countries.

A lot of countries don’t provide citizenship automatically without condition by blood. China for example, a kid only inherits citizenship if one parent is a chinese citizen and not a PR of any other country (so kids born to Chinese parents with green cards don’t count, which doesn’t really matter in this case).

Also the USA used to have weird rules about young mothers not transferring citizenship automatically (which the whole Obama birther myth relied on).

Sometimes. In many/most countries, it requires at minimum that both parents be citizens of the same country. In a few countries, dual citizenship is banned completely, so if the kid is a US citizen they cannot be the country's citizen.