The Carthaginians would've been confused by this question--dividing groups into "black" or "white" is very much a modern idea. Obviously people have noticed in the past that some people were darker than others, and some ethnic groups were darker than others, but the practice of subdividing ethnic groups into "black" and "white" is a modern phenomenon. And even today, at least in the US it's not a precise distinction because of people who are often not thought of as either, such as people from the Middle East.

...which is exactly what the Carthaginians were. Carthage was a Phoenician colony. The Phoenicians were a group originating in modern-day Lebanon, who traded across the Mediterranean and established colonies all over, including Carthage (though they did not call themselves Phoenicians, and it's unclear how exactly they grouped themselves). So, they probably looked Middle Eastern, which itself covers a variety of skin colors, between what in the US we might describe as being a somewhat swarthy white person to being olive to being brown.

If an ancient Carthaginian were to walk into a modern-day bar, we wouldn't think of them as black, but we might think of them as white either. Being a Middle Eastern person they'd look like, well, a Middle Eastern person. They would not describe themselves as "black" or "white", since those terms to describe people didn't exist the way they do today.