Why is Herodotus considered a Greek historian?

There isn't a huge amount of reliable historical data concerning Herodotus's life (somewhat ironic given that Herodotus is known as "the father of history"). Nevertheless, most historians believe that he was born in Halicarnassus, an ancient Greek city which stands in modern-day Turkey. At the time of Herodotus's birth in the fifth century BC, Halicarnassus had been culturally and ethnically Greek for close to a thousand years. The city was Greek-speaking and steeped in Greek learning. Herodotus would have imbibed such learning during his formative years, especially during his education. That said, Halicarnassus was part of the gigantic Persian Empire and would remain so for another 150 years before Alexander the Great drove the Persians out.
For the most part, the Persians were quite happy to afford their subject peoples a fair degree of autonomy in running their own affairs. The queen of Halicarnassus, Artemisia I of Caria, though herself Greek, was a loyal ally of the Persian King Xerxes I and fought alongside him against the independent Greek city-states during the second Persian invasion of Greece. Indeed, much of what we know about Artemisia comes to us from Herodotus, who praised her for her great courage and leadership skills. Despite its political subordination to the Persians, Halicarnassus under Artemisia was still able to retain a distinct Greek cultural identity, and that is why Herodotus, born and raised in the city, is considered a Greek historian.