Proof of citizenship

[Epictetus]

Yes, citizenship lists were maintained by the magistrates. I’m not sure where they were kept beforehand, but we have this from the time of Marcus Aurelius:

In the meantime, he put such safeguards about suits for personal freedom — and he was the first to do so — as to order that every citizen should bestow names upon his free-born children within thirty days after birth and declare them to the prefects of the treasury of Saturn. In the provinces, too, he established the use of public records, in which entries concerning births were to be made in the same manner as at Rome in the office of the prefects of the treasury, the purpose being that if any one born in the provinces should plead a case to prove freedom, he might submit evidence from these records. Indeed, he strengthened this entire law dealing with declarations of freedom …

Historia Augusta, Marcus Aurelius, 9.7-8

For the individual who was required to show proof of citizenship, he had a certificate.

The Roman citizen was required to register the birth of his children within thirty days before a Roman official, and he received a wooden diptych recording the declaration, which acted as a certificate of citizenship for the child for the rest of his life. Like the military diplomata this contained the names of seven witnesses, and provided a presumptive proof of citizen status. But complete validation of a claim depended in disputed case on confirmation by the witnesses that the document was a true copy of the official archive in which the enfranchisement or birth was formally registered. Similarly the enfranchisement of freedmen, which depended upon a formal act, was recorded in a documentary tabella manumissionis. Citizens of diverse origins thus came to have some form of documentary evidence of their status.

Sherwin-White, The Roman Citizenship, 316.