I can't speak to Carthaginian names (EDIT: and I realize the examples to come are from the late Republic rather than the Punic Wars), but Roman names follow a pattern of praenomen-nomen-cognomen-agnomen.
Praenomina are "first names." Gaius, Tiberius, Quintus, etc.
Nomina began as family names, like surnames in modern English, but as families got larger and larger, they stopped working to identify direct relationships and became more like "clan names." Claudius, Sempronius, Livius, etc.
Cognomina were initially personal nicknames but became attached to branches of a family with increasing regularity, acting like surnames in modern English. Names like Scipio, Caesar, Pulcher, etc. These started getting tacked on more and more, leading to the creation of the following category during the Imperial period.
Agnomina were the newest development. Technically these were just additional cognomina/nicknames, and not everyone got one of these.
It wasn't uncommon for people to have fewer names, especially if they were not born into the upper class. Pompey's birth name was Gnaeus Pompeius, for instance; just a praenomen and a nomen. He earned the cognomen "Magnus" later in life.
The reason certain Roman names became so common is that over time, fathers usually would name their first sons after themselves, with two consequences: It became difficult to tell who was who if you were discussing people in a multigenerational family, and it became difficult to stand out from your own ancestors without earning another name indicating your unique achievements.* This is where cognomina and agnomina came in: the name "Caesar," according to Gaius Iulius Caesar, came from a north-African word for "elephant," indicating (to Caesar's advantage) that an ancestor had killed an elephant in battle during the rather topical Punic Wars. To tie things together, it's worth noting that Caesar's father, Lucius Iulius Caesar, had the same nomen and cognomen, even though neither he nor his son killed an elephant during the Punic Wars over a hundred years earlier. These names had become hereditary by this time.
*(The issue of freedmen's adopting their former masters' names is a whole separate topic that wasn't hugely common during or relevant to the Punic Wars, as I recall, so I'm skipping over that.)
The consequence of these cognomina's becoming attached to a family is that the confusion between different people with similar names increased over time. Gaius Iulius Caesar's uncle was also Gaius Iulius Caesar. He got the added cognomen/agnomen "Strabo," which makes it easier to tell them apart, but the first name-surname-nickname pattern had become entrenched in that branch of the family. (Why Gaius Iulius Caesar was not given his father's praenomen, "Lucius," I don't know offhand.)
To sum up with a final example, the dictator Sulla had the full name "Lucius Cornelius Sulla Felix" upon his death. His "first name" was Lucius, but he was known to history for his cognomen, "Sulla," because it differentiated him from other members of the Cornelii. I mention this because one of Sulla's greatest rivals was named Lucius Cornelius Cinna: Same praenomen, same nomen, different cognomen (and no agnomen), and no direct relation, if I recall correctly.
A quick note: The Romans put a lot less thought into women's names. They typically only had a nomen, which was the feminine form of the woman's father's nomen. So Gaius Iulius Caesar's daughter held the full name "Iulia."
A wide variety of personal names did exist in Carthage, in fact, as we know from inscriptions. But why do we find only a small range of Carthaginian names represented in Greek and Roman sources? I can think of a few different (and not mutually exclusive) explanations:
With respect to naming practices, some Carthaginians seem to have followed the principle of papponymy, i.e., naming a son after his grandfather, which would have led to the repetition of certain names across alternating generations. This probably accounts for Hannibal Barca's name, since his grandfather was also named Hannibal.
Names such as Hannibal (from Punic Ḥnbʿl), Hamilcar (possibly from ʿbdmlqrt or Ḥmlk), and Hasdrubal (ʿzrbʿl) were simply very popular. Indeed, one study counts 312 attestations of Ḥnbʿl, 754 of ʿbdmlqrt, 224 of Ḥmlk, and 411 of ʿzrbʿl, in contrast to, say, just four attestations of Eshmunazar (ʾšmnʿzr).
Many Greeks and Romans, I suspect, had difficulty in remembering or representing Phoenician-Punic names, and this may have led to the privileging of more familiar or easily rendered ones like Mago, Hanno, or Hannibal in our literary sources. Note that other names often get mangled or condensed in their Greek and Latin forms: Bodashtart (bdʿštrt) becomes Βώσταρ, Bostar, etc.; Bodmelqart (Bdmlqrt) becomes Βώνχαρ, Boncar, etc.; Gersakun (Grskn) becomes Γέσκων, Gisco, etc.; and Melqarthilles (Mlqrtḥlṣ) becomes Καρθάλων or Carthalo.
Sources:
Benz, Frank L. Personal Names in the Phoenician and Punic Inscriptions. Studia Pohl 8. Rome: Biblical Institute Press, 1972.
Ferjaoui, Ahmed. Recherches sur les relations entre l'Orient phénicien et Carthage. Orbis Biblicus et Orientalis 124. Fribourg: Editions Universitaires Fribourg Suisse; Göttingen: Vandenhoeck & Ruprecht, 1993.
Geus, Klaus. Prosopographie der literarisch bezeugten Karthager. Orientalia Lovaniensia Analecta 59, Studia Phoenicia 13. Leuven: Peeters, 1994.
Halff, Giselle. "L'onomastique punique de Carthage: répertoire et commentaire." Karthago 12 (1963-1964): 61-146.