Most Greek city states kept citizenship lists, as well as other lists, such as for the metics, non citizens who had to pay a tax. In Athens, much of the record keeping was done on the level of the demes, where the citizen would be enrolled at the age of 18 ([Aristotle] Athenaion Politeia 42.1), though there were also other lists that citizens kept based on who is eligible or liable for what.

Of course, citizenship is not static. Using the Athenian example, we know of multiple instances where the qualifications were changed or the rolls were purged. After the end of the Peisistratid tyranny, [Aristotle] says that the Athenians purged the rolls of three thousand people who had illegitimately received citizenship (Ath.Pol. 13.5).

Most famously, Pericles enacted a citizenship law in Athens during the fifth century, restricting citizenship only to those people who were born of two citizen parents ([Aristotle] Ath.Pol. 26.3–4; Plutarch, Pericles 37.3). What exactly this law did is somewhat contentious, with suggestions ranging from there being too many mixed marriages, to restricting citizenship privileges, to ethnic exclusivity. It is even possible that because the citizenship rolls could be populated from the deme-up, if the deme vouched for someone as a citizen, then you were. Lingering over all of this in the Athenian example is that the Assembly could bestow citizenship as an honor, and numerous inscriptions testify to this.

To the question of investigations into fraud, the most reliable arbiter of this in Athens was through the courts. We actually have one court speech, [Demosthenes] 59, Against Neaera ( probably by a man named Apollodorus) that talks about a case where the prosecutor alleges that Neaera has falsely presented herself as an Athenian citizen in order to secure citizenship for her children. In the speech, Apollodorus tells the jury "if found guilty, he shall be sold [into slavery] along with his property, and a third of the proceeds shall go to the victorious prosecutor. The same holds for a woman" (59.16).

Apollodorus also says that anyone had the right to bring charges against the person suspected of not being a citizen.

Other systems existed outside of Athens, but the combination of citizenship as an honor that could be extended and official lists were likely fairly ubiquitous. As was the connection between citizenship and the law. In perhaps the most famous instance of this, when he was arrested Paul of Tarsus asserted his right to be tried before Caesar on the grounds that he was a Roman Citizen (Acts 22:25–29). Whether he did or not is another issue, but the claim evidently worked in a moment when the Centurion holding him didn't have a list handy and the potential risk of illegally violating a Roman citizen's rights outweighed the reward of holding a suspected criminal.

By 212, the emperor Caracalla pushed a new edict making every free man within the Roman empire a citizen. Cassius Dio says that the main purpose of the change was tax revenue (78.9).

tl;dr Absolutely the people of the ancient Mediterranean kept lists of citizens, but the lists and who qualified were subject to change for a host of reasons. In Athens, fraud detection amounted to a legal proceeding.