I assume you are referring to the modern day countries of Macedonia and Greece?

Current day Macedonia is mostly comprised of territory which was not part of the ancient Macedonia which Alexander belonged to. (Most of what was ancient Macedonia resides in Greece). In fact this is a huge point of contention between Macedonia and Greece. The Greeks feel as if the Macedonians are co-opting their heritage. This is further complicated by the fact that modern day Macedonians are ethnically Slavic, and arrived in the region roughly 1000 years after Alexander the Great's reign, so they are not related to the Macedonians of the Classical period.

As another user has said, Alexander himself was taught by Aristotle, a prominent Greek. He spoke fluent Greek, and united the various city states to his cause before marching against the Persian Empire.

Although the land comprising of ancient Macedon is now a part of modern Greece in Classical times Macedonians were generally considered barbarians, ie non-Hellenes, by their southern neighbors. The Macedonian royal family, however, admired and promoted Greek high culture and believed themselves to be descended from the hero Heracles, through his great-grandson Temenus. Macedonians belonging to the royal Argaed dynasty were allowed to compete in the Olympic games while common Macedonians were not. As only Greeks were permitted to enter the games this indicates that Macedonian kings like Alexander were considered Hellenes. Furthermore, both he and his father Phillip portrayed themselves as champions of the Greek city-states, uniting them in the face of their ancient enemy - Persia. Claiming descent from the heroes Perseus, Achilles and Heracles, and being a deep admirer of Classical Greek culture, Alexander the Great no doubt considered himself a Greek while many of the peoples he encountered during his conquests, unfamiliar with the discrepancies between Greek and Macedonian, would have considered him such as well.

The independent republic established in 1991 adopted the name of its autonomous Yugoslav predecessor, in turn inherited from the name used internationally for a broader region including modern Greek Macedonia and a smaller area incorporated in modern Bulgaria.

I recall when the kingdom of Philip and Alexander was termed "Macedon" to differentiate it from the later broad geographical area. It's regrettable that so much passion should be spent squabbling over a name when there are two available of which the unused one formerly commanded higher status in connection with the disputed period.