True, they considered themselves Phoenician; the Byzantines, Ottomans, and Russians all considered themselves Roman as well. Well for ethnic identity that might be fine, for sorting out historical cultures I think it's unhelpful. They may have still spoken a dialect of Phoenician and worshiped Phoenician gods, but culturally the Carthaginians were very different from their Canaanite forebears. I wouldn't call them Hellenized per se, but they were certainly influenced by Hellenic culture, as well as Berber and other cultures.
Russians considered themselves Roman? That's somehow worse than calling a Germanic Kingdom the "Holy Roman Empire" (A more accurate title would have been "The Not-Holy Not-Roman Not-Empire"). But in the case of the Carthaginians, it actually makes sense that they would think of themselves as Phoenicians. They kinda were.
As for being influenced by Hellenistic culture, no doubt Phoenicia was as well, long before they founded Carthage, being that they had long been trading with the Greeks, and would have had a lot of contact with them. Each Phoenician colony would likely be influenced by local peoples, so it's unsurprising that the Carthaginians were influenced by Berbers. Australia, New Zealand, South Africa, Canada...All of these were influenced by the indigenous peoples of their regions, but for a time I'm sure all the colonists (and convicts) considered themselves British/Irish. They all remained part of Greater Britain anyway for many years, and if they had have remained dependent colonies, they would still be to this day. It was independence from Britain that made them officially other countries. Otherwise they might have been like what Greenland is to Denmark...Culturally (and in most cases ethnically) very different, but still "Danes". Can you imagine Firaxis adding Greenland as a separate civ to Denmark? I can't.
Why not Dido? I went over this elsewhere, but I'll reiterate it here. She is of very dubious historicity. She's attested only in Greek and Roman sources. And she's hardly a big personality compared to the Magonids, Barcids, and other later rulers of Carthage.
That's true...fair enough. Somehow I imagine Firaxis will probably have her as the leader anyway, as she has appeared before, and she is rather iconic. I think that the main issue is that we don't no anything about her from non-Greco-Roman sources.