This was touched on a little in a previous discussion on proving Roman citizenship.

The AutoMod reminds me to mention u/LegalAction and u/mythoplokos in that thread.

Paul of Tarsus's claim was particularly tricky because he received his citizenship by virtue of where he was born. Tarsus was a Greek speaking city-state on the south coast of Turkey. Tarsus was never conquered, per se, but swore fealty to Rome in exchange for all its free inhabitants receiving Roman citizenship. This type of citizenship, stemming from the place of birth, is known as Ius Soli (or Jus Soli), as opposed to Ius Sangui which is citizenship derived from blood.

The entire New Testament appeared to be written in Greek, and it is very possible Paul grew up speaking Greek as his mother tongue, Hebrew as a liturgical language, and learned Aramaic when he moved to the Levant later in life. Paul's claimed citizenship being Ius Soli, there would be no paper trail that Paul would be required to provide.

If Paul spoke Greek like someone from Tarsus, or just appeared credible, it may be that caused the Magistrate to believe him. Paul may have just been bluffing, having been from Athens, but the Magistrate did not want to take the risk of getting himself in trouble for the Jews who were crying for Paul to be punished. However, what the Magistrate did to confirm Paul's story would be speculation since our only record of it is the Book of Acts. Paul specifically knew his rights as a citizenship and pled them specifically according to Acts.

The important thing to note is his particular claim to Ius Soli citizenship was impossible to prove or disprove and put the magistrate at risk if Paul's family showed up later to corroborate and appealed to Rome to have the magistrate punished.