Actually, in Herodotus’ The Histories, there is a story about Alexander wishing to compete in the Olympic games (which were only open to Greeks.) In order to compete, Alexander had prove Greek ancestry. He was able to do so and was thereafter allowed to participate in the games.
The Greeks of Alexander the Great's time did not think much of him. Thebes rebelled and he had the city destroyed. The Athenians were generally hostile. Demosthenes considered the Macedonians to be barbarians. Sparta fought a war against Macedonia while Alexander was off fighting the Persians.
The original Macedonia was smaller until Philip II conquered more territory including what is now the Republic of Macedonia. The inhabitants of the Republic are mostly Slavic-speaking (Macedonian is very similar to Bulgarian), with an Albanian-speaking minority. But they are probably partly descended from the ancient inhabitants of the region--it is unlikely that the Slavs exterminated everyone who was there when they arrived.
That Greece (10 million people, member of NATO and the EU) has anything to fear from Macedonia (2 million people) is ludicrous. I don't see why anyone wants to claim Alexander the Great. He conquered territory and got a lot of people killed. So did Genghis Khan. So did Tamerlane. So did Napoleon.