Hi there..

I'll be happy to provide sources if needed, but according to what I've studied on Carthage / Phoenician history (quite a lot), the makeup of Carthage was primarily those with ties to the mother city of Tyre and/or other Phoenician cities.

It's slightly broader than that though...so let me just do a quick background..

So you have Tyre and other Phoenician cities, all sharing a common language, religion, commerce system, etc. (despite the fact they maintained independence from each other). So naturally when Tyre the largest and wealthiest began to be pressured by neighboring forces, they looked to their western Mediterranean colonies to move. Many people would stay in Tyre, but you had an exodus of Phoenicians from multiple cities leaving to the newly founded Carthage, the economic hub of the Mediterranean. Like most Phoenicians, they all championed their culture, but revered their home cities. So you had mostly Tyrians arriving, making up the bulk of the city's populace / hierarchy. This would mean most future Carthaginians associated Tyre as Phoenicia's "mother city", and their culture's roots. Exactly how or when Carthaginians separated themselves from their Phoenician ancestors is hard to say, but slight changes in religious worship, hierarchy, public action / belief occurred over time, and as Phoenician cities declined in economic independence and trade primarily moved to Carthage, pretty soon you had this economic beast sitting in the middle of the Mediterranean.

That's the basic outline...lol..so anyway..

As I said you had citizens with strong ties to their homeland, so naturally they elected only officials/ suffetes / elders / generals with roots of their own only. What Hannibal or his family was is unknown (as far as I know), but they most likely associated themselves with Tyre (despite the fact that they had large issues with higher-ups in Carthage). I know when Hannibal fled Roman occupied Carthage, he went to Tyre at first.

Carthaginians were proud of their religious ties to Phoenicia (and subsequently it's colonies) and made pilgrimages to several sites including Gades /Cadiz, Tyre, Byblos.

Here is where most people get mixed up...

Carthage obviously employed the use of Mercenaries, of a multitude of ethnicities. Did some of these cultures live within Carthage? Yes. Did they hold power? No. Not at least one of very high stature. There were periods in Carthage when foreigners were offered places to live and many moved in. However, they were asked to leave while religious ceremonies took place within the city. Also, they were never voted in as Generals of CARTHAGINIAN forces, or any other real higher office. You would see Generals whom commanded men of their own ethnicity, but they in turn responded to a Carthaginian commander-in-chief (so to speak). Despite the fact it was a Phoenician born only system, being a higher up in Carthage had its pros and DEFINITE cons. Generals whom lost frequently were crucified (fact), and in times of desperation the act of child sacrifice DID happen (let me stress DESPERATION), and when it did happen, it was the children of the elite class whom were used.

Despite the fact you did have a general mixing of cultures within Carthage, it seems the Berbers and Numidians especially maintained their own kingdoms. It was the Numidians whom betrayed Hannibal in his fateful battle with Scipio on Carthaginian soil.

I wasn't planning on writing that much...sorry. I also apologize if some of that is not clear. It's pretty much fired off the hip lol. Hope it helps!