The short answer is: we're not sure, but it's likely Thales came from a Phoenician family that migrated to Miletus. As far as I know, Herodotus is the earliest source we have that talks about Thales. And even this source is written about a hundred years after Thales had already died. So we have very little to base a clear answer to this question.

Now for the more interesting long answer. First it must be mentioned that the Milesians in the time of Thales did not call or consider themselves first and foremost as 'Greek', they would call themselves Milesians first, and Ionians second. At the same time, the people we now call Phoenicians, also never called themselves that. Second, it's important to understand that ethnic categories such as Greek, Phoenician, Carian were in reality not as clear-cut as they seem. The area around Miletus was inhabited since the neolithic, thousands of years before there were Greeks or Phoenicians. Already during the Bronze Age and later Archaic Age, Miletus was a powerful regional city with a rich history. The people who lived there were probably a changing mixture of people we would now call Lydians, Myceneans, Minoans, Carians, Phoenicians, Greeks, ...

So how can we determine if someone was Greek or Phoenician (keeping in mind that these are categories of a later date and not used by the historical people we're talking about)?

First off, language. Miletus was part of the Ionian League. This was a defensive/religious alliance between twelve independent city-states on the western coast of what is now Turkey. They (or at least the elites of these cities) spoke the Ionian dialect of Greek. But even between the cities of the Ionian League there were many differences in dialect. Ionian Greek was also spoken in Athens, and many Ionians had the notion that they were the descendants of Athenians that migrated across the Aegean Sea. But this idea is not to be taken too literal and mostly a result of Athenian expansion in the centuries after Thales. The Athenians tried to expand their power and a semi-legendary common origin was a successful way of making alliances.

That brings us to population and migrations. The Ionian migration into the region probably occurred around the 11th century BCE. These people presumably came mostly from Attica and Boeotia. The way in which they mingled and lived together with the existing populations is uncertain and most likely happened in different ways each time according to the circumstances. In other instances of Greek migrations we see that they could marry into the local families, they could live together on relatively equal footing, they could go to war and chase off or enslave the people, and everything in between. In whatever way it happened, in these twelve cities, the Greek culture became the dominant culture during the following centuries.

Miletus, like the other city-states of the Ionian League, had a political organisation based on different tribes. It seems there were six tribes. Four of them would have their origin in Greece and two would be local. How large these groups were, or what differences in social standing they had is unfortunately not clearly known. Nevertheless, this shows us that, even though Milesians would later consider themselves Greek (or at least Ionian), their origins are very murky and any idea about ethnic heterogeneity was mostly an ideal not at all reflected in reality.

About a century before the birth of Thales, Greek pottery entered a phase called the Orientalizing Period. During this time many cultural influences from the eastern mediterranean entered the Greek speaking communities. It is very likely that a major driving force of these cultural developments were traders and craftsmen who originated in the Phoenician city-states such as Tyre, Sidon, Byblos and had contacts with Greek cities and towns. At first, Phoenician traders would visit and conduct business in these towns (and Miletus was a very important crossroads of the mediterranean trade). Later, these traders would settle down and create small industries and trade hubs. There's no question these traders and craftsmen were at first looked at as foreigners, but they could amass considerable wealth and were probably at times able to secure citizenship and an important place in their new homes.

So where does that leave us with Thales? If we look at the language, Thales was Greek. For most Greeks, language was the first and most important thing that distinguishes Greeks from barbarians. Since Thales lived in a city dominated and inhabited mostly by people who spoke Ionian Greek, he would have spoken this language as well.

That Thales was of Phoenician origin, as Herodotus and others mention, is also possible. Lots of Phoenicians spread out across the mediterranean in the centuries before Thales. As traders and craftsmen, some of them were able to become quite rich and secure citizenship and prominent places in the societies they migrated to. That Thales is a descendent of Phoenician migrants is therefore perfectly imaginable.

In conclusion: was Thales Greek of Phoenician? The answer is most likely: both, with countless caveats about the complexity of ancient ethnicity and identity.

Main sources:

Roebuck, C. 'Tribal Organization in Ionia' (1961).

Greaves, A. M. 'The Land of Ionia' (2010).

And of course the Histories of Herodotus.

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