For me, the general process would require a police report for lost/stolen ID (mandatory, so that it can be marked as lost/stolen so that it would be detected if someone tries to use it) and verification with the data they have on file - nowadays with EU biometric IDs they can be quite sure that I'm the same person as the one who got the previous ID as the face and fingerprints can be verified.

There's an honor system in many places. You sign a document stating you are who you say you are, and have it witnessed by someone who is "deemed trustworthy" - local police, teacher, clergy.