We tend to know quite a bit about the names of citizens of Carthage, although they likely came from middle to upper classes. First, we know a lot of names from the tophet, because people would write their names on the votive stelae. Second, after the Second Punic war Hannibal began a political campaign, part of which was empowering the common people. We have a few dedications, such as the Avenue de la Republique inscription http://imgur.com/a/ashmT which list the names of the people who built the road. The inscription reads:

“… this street was opened and constructed, leading to the New Gate square, which is in the southern wall(?), by the people of Carthage, in the year of the suffetes Safat and Adonibaal, in the time of the magistrature(?) of Adonibaal, son of Esmounhilles(?) son of B(… and of … son of Bodmel)qart, son of Hanno and their colleagues. Those in charge of the work (were): ‘Abdelmelqart, son of Baalhanno, son of Bodmelqart, as road engineer, Yehawwielon the brother (of Bodmelqart, as quarrymen[?]). (And there worked on it all) the tradesmen, porters, packers(?) from the plain of the town, the weighers of small change(?) and (those) who have neither (silver[?] nor gold[?] and also those who do, the gold smelters, the craftsmen who make vessels(?), and (those employed) in working the furnaces, and the makers of sandals(?). (all) together. And (if anyone effaces this inscription) our accountants will punish that man with a fine of one thousand (shekels of) silver, over and above (X) measures(?), (for the price of the inscription [?])

One interesting trend in naming is how people have gods in their name. The terms "Adon" and "baal" both mean lord, and in Carthage usually referred to Baal Hammon, the most important god of Carthage. Esmoun was a god of healing, very similar to the Greek god Aesculpius. Melqart was at one point the most important god of Carthage, but fell in prominence as relationships with Tyre declined. The other half of each name describes the relationship to the god. The name Hannibal means "he who enjoys Baal's favor" and Hasdrubal means "He who has Baal's help" and Abdelmelqart means "He who is pledged to the service of Melqart".

Romans kept records of civilians with citizenships. And conducted a census in the Campus Martius every 5 years at Rome (not sure about other cities), but yes. To keep track of civilians with citizenship grants a record was kept. Not sure where or if it survives today, it was a first in human history.