It's possible, I believe, that Carthage could have risen to control the Mediterranean (as Rome did), but Carthage would have had to change. That change might have come from the influence of the victorious Barca family and their supporters. Necessary changes would have includied expanding citizenship to other Phoenicians in North Africa, even a limited form of citizenship.
Carthage would have eventually been overwhelmed by some Mediterranean power if the Romans had not annihilated them. They may have lasted longer, spread farther, and had a greater effect on the Mediterranean basin, but eventually they would collapse. A good example of a Carthaginian like collapse would be the earlier Athenian Empire. The Athenian Empire was based on trade and naval power, and had colonies around the Aegean, as well as farther out into the Mediterranean. The Athenian Empire was not defeated in war, but because of it. Athens had a mercenary fleet and paid allies to field land forces, much like the Carthaginians did with the Numidians and Spaniards. After fighting the Peloponnesian league to a standstill, the Athenian treasury ran out and the city-state collapsed internally. If the Carthaginians had managed to defeat Rome, or confined the Romans to Italy, they would still have been unable to cope with the military expenses, unlike the Romans who had citizen soldiers. Victory over the Italians would simply have given the Carthaginians another foe, perhaps the Gauls, or one of the Diadochi kingdoms in the Eastern Mediterranean.