Consider a person in the US which has no papers but claiming he has a US citizenship. How the authorities verify that he is indeed a citizen rather than an illegal immigrant, given that neither passport nor residency registration is mandatory?
Ultimately birth records, represented by a birth certificate.
This is just an official document that proves a certain person is a US citizen. The question conflates this with how a person proves that they are who they claim they are.
For that purpose, there are all kinds of databases (which do include photos) kept by various agencies. If you try using someone else's identity (aka identiy theft), you'll eventually run into a situation where someone notices that the photos don't match. Or, if you look similar enough, someone will notice that two people are using the same identity. You can't use a dead person's identity either because deaths are also recorded in these databases.
For Americans born in the US, a birth certificate (typically issued by the U.S. State where the person was born, or the State Department if the person was born abroad), possibly certified, usually suffices (see below for limitations when applying for a passport). For others, certificate of Citizenship or Naturalization Certificate (or a U.S. passport). Reference here.