There are probably folks more qualified than I to answer this, but I did recently struggle with this exact question after reading In Search of the Phoenicians by Josephine Quinn, and I've attended one of her Oxford lectures, so hopefully I'm not too inexpert by the standards of the sub, because I do think that Quinn's book is a great starting point for this question and if nothing else I wanted to point you in its direction.

As a starting point, Maria Aubet includes a reference to the complex identity of the Phoenicians as part of the definitions section in her introduction to Phoenicians and the West; she then goes on to provide one of the earliest English-language summaries of the archaeological evidence we have of Phoenician settlements in Iberia. Quinn, by contrast, focuses on what we know about Phoenician identity as such, through the lens of contemporary ideas about identity.

It turns out that various ancient cultures probably didn’t actually self-identify as being part of the culture we assign to them, and the book gave me a lot of useful paradigms.

Where different identities come from is an important consideration, especially when thinking about questions of authenticity and imperialism. Historically, I’ve tended to default to the idea that most cultures have firm enough “edges” that we can tell where they begin and end. I’m used to regions being defined by geography, with language and clothing choices and other cultural markers being fairly consistent within those regions: the British Isles, the Appalachian Mountains, the Greek mountains, Silicon Valley, the Basque people of the Pyrenees mountains, the Inca in the Andes, etc.

The Ancient Greeks weren’t united by any stretch of the imagination, though, and there’s a lot of debatae surrounding just how contiguous Chinese history really is. Athenians saw themselves as part of the Hellenic world. Tyrants of Syracuse by Jeff Champion, for example, makes it pretty clear that they saw non-Greeks as frankly barbaric, even as they fought with rival Greek city-states. But even Greek identities are pretty complex;

in relation to “Hellenicity,” for example, scholars have delineated a shift in the fifth century BCE from an “aggregative” conception of Greek identity founded largely on shared history and traditions to a somewhat more oppositional approach based on distinction from non-Greeks, especially Persians, and then another in the fourth century BCE, when Greek intellectuals themselves debated whether Greekness should be based on a shared past or on shared culture and values in the contemporary world. By the Hellenistic period, at least in Egypt, the term “Hellene” (Greek) was in official documents simply an indication of a privileged tax status, and those so labeled could be Jews, Thracians—or, indeed, Egyptians. (Quinn, 2017)

Basically, the Hellenic identity shifted over time, insofar as it was ever clearly defined — a fact debated by the ancient Greeks themselves!

Why did Greek authors recognize some sailors from other Mediterranean city-states as fellow-Greeks and others as something else? One thing that would have clearly distinguished “Phoenicians” from Greeks for Greeks was their language. The role of the Greek language as a criterion for Greekness itself is disputed: Jonathan Hall has argued that despite a great deal of evidence for mutual comprehension across dialects, there is little for the “awareness of a common Hellenic language” before the fifth century, and that even then the relative linguistic homogeneity of the epigraphical evidence from the Classical period might conceal a great “diversity of oral idioms.” Nonetheless, it would not be necessary for a Greek to be able to understand every other form of spoken Greek to note that Phoenician is very different from all of them: one can imagine a Londoner baffled by a Glaswegian, but still readily able to distinguish the language being spoken from Arabic.

Identity is messy. Ethnic, religious, social and political meanings can get caught up in the same terminology.

Recent studies have shown that such familiar groups as the Celts of ancient Britain and Ireland and the Minoans of ancient Crete were essentially invented in the modern period by the archaeologists who first studied or “discovered” them, and even the collective identity of the Greeks can be called into question. (Quinn, 2017)

Being honest, I quite can’t tell if Quinn is taking the position that these identities are purely modern invention, or mostly saying that we don’t know enough and shouldn’t assume. Taxonomies are useful, but we need to bear in mind that the ancients used different taxonomies to define themselves.

So, for example, the Phoenicians and the Canaanites are often conflated, since they occupied roughly the same spot in the Levant. The difference between them seems to largely come down to who is doing the defining. The term Canaanites seems to have been coined by the ancient Israelites to broadly mean “enemies of Israel” (of which there were a lot), and Israel consisted mostly of inland city-states.

The term Phoenicia, by contrast, comes to us from the Greeks, who were mostly interacting with city-states like Tyre and Sidon from the sea. The Greeks and later the Romans are basically the only ones who actually label Phoenicia as such, instead of referring to the city-states individually or referencing the broader Levantine area via some other term, like “Hittite.” Although various historical figures did give geographic locations for Phoenicia, from the way it seemed like the term got used, “Phoenician” almost seems more synonymous with “merchant” or “sailor from the coastal cities” than “person from one of this handful of city-states or their surrounding lands, who speaks a particular language.”

Although Phoenician identity is pretty hard to define, it seems like the Punic world is a bit easier to trace. “Punic” is the (Latin) term used to refer to the people of Carthage and the areas under its influence, i.e. the Phoenician colonies in the western Mediterranean region. Since Carthage was a “rising Imperial power” it went out of its way to forge an identity for itself. This isn’t a unique phenomenon: Alexander the Great did something similar during the fight against Persia, even though his own identity as a Greek was somewhat shaky.

Quinn also says,

Understanding the ancient world as divided into a series of “peoples,” each with what Anthony Smith called “a distinctive shared culture,” depends on an implausible idea of how culture works.

The whole idea behind cultural diffusion is that “culture” is more of a spectrum than a quilt. It’s useful to mark boundaries between groups when it makes sense to do so, but it’s equally important to remember that real life is messy and sometimes boundaries are artificially imposed. Periods of conflict, whether violent or economic, make a good example of when people might differentiate between themselves and “the other.” But those identities are also messy and flexible.

(Note: if this answer seems slightly disjointed, I'm sorry, I actually adapted it from an article I wrote for fantasy authors back in September. I did my best to "smooth it out" but I might have missed something 😅)