Only some of Iberia was under carthaginian rule, and even there very briefly. Before Hamilcar, it was only a few Phoenicians trading posts. After the second punic war, they were gone. 1 generation is not really enough to forge a local culture. I'd always imagined the Carthaginians as occupiers, who may or may not get along with the local population, but I would think it a stretch to think there was a real Iberian-Punic culture outside a few trading centres.
The Phoenician presence in the Iberian Peninsula is quite old, at least from the 9th century BC, but it was mostly coastal, with strong trade activity, and small colonies.
When the Carthaginians arrived they already knew that Phoenician trade network, but the relation between the Carthaginians and the rest of the Phoenician colonies wasn’t always peaceful, the colonies were independent and probably pretended to be that way, two examples of that, more or less known are Utica (in Africa) and Gades (in Iberia). Gades quite soon was able to liberate from the Carthaginian domain and was a relevant Roman ally.
A strong potentate in the Southwest of the Iberian Peninsula was Tartessos (that can be or not the Biblical Tarshish), and there is a myth that the Cartheginians destroyed them.
In the South and Southeast of the Peninsula we had the Iberians, that quite soon suffered influences of the commerce with the Orient (Greece, Egypt, Cyprus, Phoenicia…), and that also established relations with the Carthaginians.
So, in its peak I see the Carthaginian “domain” in the South and Southeast as proto-feudal and vassalic, and indirect rule over a set of tributary tribes, that contributed with taxes and armed men for the Carthaginian campaigns, and begun to be even more integrated in the Phoenician trade network.
As stated they only really ruled the coastal areas - though Hannibal did once go on a long and very bloody expedition to the far north at one point.
They were extremely parasitic- using the land for soldiers, resources, food etc whilst giving nothing in return. Incredibly high-handed in a way that led to readymade support for the Romans when they arrived. Not entirely unlike the Visigoths germinating ready made support for islamic invaders in 711.
The death of Hasdrubal the Fair seems witness to their attitude