Greeks called them - Phoenicians
Romans called them - Punics

What did they call themselves?

They most likely would've identified first with their city state but acknowledged they were all part of common overarching culture, language, religion they called Kena'ani (Canaanite).

..similar to how Spartans and Athenians would've identified with their city first but acknowledged they were all Hellenic peoples.

The Iliad even breifly mentions Sidonians and Tyrians separately, by those names, without connecting the two to each other.

Carthage, the most famous Phoenician colony, was a diverse city with a ruling class that identified as Kena'ani & kept strong ties with their mother city of Tyre.

Tyre or as they called in their language: "Sūr" -pronounced "Soorr".

Carthagians right up until their demise refered to called themselves "Bin Sūr" ie "Sons of Tyre".
And would send a tithe of their city's annual earnings, back to the Temple of Melqart in Tyre.
Tyrians under Persian rule, refused to orders to attack Carthage their daughter city..

..Many Tyrians were evacuated to Carthage before Alexander laid seige to the city.
Hannibal fled to Tyre first and then Antioch after his defeat at Zama, before offering his services to the Seleucids.