Alexander the Great likely would have called himself a Greek. The Ancient Makedonians had Greek style names, worshiped the Greek Gods and followed Greek culture and customs. Alexander himself was educated by Aristoteles.
Their conquest of Greece was in part motivated by their desire to be seen as one of the great cities of Greece, like Athens, Thebes and Sparta.
Once he had conquered a large part of his Empire Alexander actively promoted Greek culture across his lands, including a desire to merge Greek and Persian cultures. It was partially due to this that Hellenism ermerged and Greek culture started to dominate the Mediterranean sphere (especially in the East) until Rome showed up.
The Ancient Makedonians have nothing to do with the modern, Slavic Republic of Macedonia. The ancestors of modern Slavs lived somewhere else during that time.
Didn't his father deliberately call himself Philip of Macedon ?
Troy was not Greek and is fictional. The Troyan war possibly has some influence from a war that destroyed the Hittite City of Wilusa.