YSK: How to document that you are a US citizen
https://www.usa.gov/prove-us-citizenship
The most common way to show that you are a US citizen is to show a passport. However, only about 50% of US citizens have a passport.
An alternative to a passport is a Certificate of Citizenship.
Certificates of Citizenship and Naturalization show proof that someone is a U.S. citizen. The website https://www.usa.gov/prove-us-citizenship shows how you can get or replace these documents.
Prove your citizenship: born in the U.S. with no birth certificate. If you were born in the U.S. and have no birth certificate, learn how to get documentation to prove you are a U.S. citizen.
Prove your citizenship: born outside the U.S. to a U.S. citizen parent. Prove your U.S. citizenship without a birth certificate if you were born outside the U.S. to a U.S. citizen.
No, the closest thing to an ID card in the US is either a drivers license (which does not prove citizenship, any legal resident can get one) or a Passport (which does require citizenship by review of birth certificate).
Any US citizen who does not have a Passport (or Certificate of Citizenship/Naturalization) has no document proving their citizenship. A very high number of US citizens have no documentation to show their status as citizens.