Before starting, it is important to note that our understanding of this period in Carthaginian history is sketchy at best. They left no surviving written records behind and very few epigraphic documents.

Given that, it seems from what we can gather that Carthage employed a variety of different arrangements with various her various "territories". Some cities, like Gadir (Cadiz) and Utica, were powerful in and of themselves and seemed to remain largely independent both in their governance and in their foreign policy. They must have acknowledged Carthage hegemony in some way, probably in taxes/tribute and unequal or preferential trade arrangements. The Romans had similar arrangements with powerful or influential cities, a great example being Massilia (Marseilles), which remained "independent" inside the Roman Province until 48 BCE.

Other Punic enclaves had fewer rights. This seems to have been the case with Lepcis Magna, which did not have independent control over its foreign policy. Motya also seems to have been in this category.

Other places were directly controlled. Kerkouanne (on Cap Bon) was akin to what the Romans would call a colonia, with archaeology which is not only distinctly "Punic" but also distinctly "Carthaginian." The city also successively pushed into the Tunisia hinterland to secure agricultural land, though this never seemed to have been their main aim. They were never a true territorial empire like the Romans and often did not expand their direct territory after defeating their neighbors in war (like the Numidians). We hear that Carthage took over the old Phoenician maritime "emporia" in Spain and revamped them, and they also founded new enclaves (like in Ibiza). We hear about these in the treaty with Rome in 509 BCE, which suggests that even by that early date, they were securing overseas possessions.

And this makes perfect sense. Carthaginian hegemonic leverage was in ships, not infantry. They compelled maritime cities to submit to them by dominating them on the sea, by restricting their trade opportunities, or by outright blockade. A major facet of Punic economy was the cultivation of the murex (a type of sea creature), from which they extracted the precious purple dye. They built artificial pens out on the coast to farm the murex, meaning that most Punic cities tended to be on or near the coast. This put them under the Carthaginian thumb by default. Carthage was already the largest and richest city by 750 BCE and only increased her marketshare thereafter. More money coming in means more ships, which means more money coming in, ... In this way Carthage was very similar to Athens in the "Delian League," and Carthage probably behaved in a similar way: using a fleet advantage to leverage weaker cities into disadvantaged situations through time until they became vassals/dependents.

Like I said before, their system was probably very similar to the Romans: a range of idiosyncratic "arrangements" between Carthage and dependent cities based on the circumstances in each case. The negotiating chips were political autonomy, freedom of foreign policy, trade agreements and currency dependence, conscription obligations (men or ships), preferential treatment for Carthaginian shipping in port as far as taxes/dues, military "protection," rights of intermarriage, and plain ol' tribute. We have examples of Carthage intervening to protect the interests of other formerly Phoenician enclaves, like Lepcis. That city had been founded by 700 at least, and was therefore nearly as old as Carthage, but very quickly became under the Carthaginian thumb. When the Greeks showed up a day's journey to the east to form a new colony, Carthage worked in concert with the locals (the Macae) the drive them away. This was surely to protect the economic and territorial interests of Lepcis.

In conscription rights, in particular, Carthage seems to have differed from the Roman system, the latter of which depended on "allied city" levies to fill out infantry armies. Carthage always depended on mercenaries to a larger degree, and this probably makes sense given that Carthage had a lot more cash flow than Rome in the 5th-2nd centuries BCE.

Carthage was also much more jealous with citizen rights. I don't know if inhabitants of Gadir and Utica were citizens, but Carthage did not seem to create seed colonies of citizens abroad, or at least not in large numbers. Punic cities did enjoy varying levels of non-citizen status, just like Roman municipia, but it is not clear exactly what that meant. No one really knows exactly what it meant to be a citizen of Carthage, though most agree that it was driven by wealth as much as birth. The city was a Republic more than a monarchy (the mother cities had kings), though, like Rome, probably a very narrow oligarchy which was often dominated by a single aristocrat and his posse. There was a senate (a gerousia, or council of elders) which was very similar to Rome's in function. There were also apparently "boards" which achieved various tasks, what Aristotle called pentarchies (boards of five members). Again, very similar to Rome's endless boards and committees.

For the best and most recent academic sources on Carthage:

S. Lancel, Carthage (1995). D. Hoyoes, The Carthaginians (2010).