There is no evidence to say he wasn’t Greek, apart from rumours about his mum being half barbaric, and him having non Greeks in his army along with Greeks, but that’s not very uncommon in armies.
He was Greek, if any of the “Greek” heroes turn out not to be Greek it would probably most likely be Achilles and not Alexander the Great any smart Albanian who’s studied Alexander the Great knows he was Greek and not Illyrian, unless his mother really was half Illyrian, that still makes him a Greek with Illyrian ancestry, it doesn’t take away his Greek origins.
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First of all, the issue is more complex and simple than we make it to be in the same time. All these tribes who lived next to each other were very similar to identical, and for sure Proto-Hellenes were brothers/cousins of the exact same stock as Proto-Illyrians and Thracians.
What made them different after 1000 years? The amount of local intermixing, assuming the Hellenes inherited far more local genes than the more Northern Illyrians and Thracians were the population could have been more scarce, but also the further South (or any direction) the more diluted would have been the blood of the tribes, e.g. the Hellenes didn't come overnight from North Eastern Europe to Southern Greece.
So considering this, personally I see an obvious connection between the local unknown culture of Greece and its appropriation by the incoming Hellenes. For sure their forefathers who dwelt more North were identical to the Illyrians and Thracians in their appearance, way of life, etc. The caste system of India comes to mind with such an example.
Now back on the topic, all those small tribes around the area of South Illyria, Paeonia, Epirus, Makedonia, Emathia, Mygdonia, Orestis, etc. were a bunch of highlanders cousins to each other. What fate brought for them later on is different depending on the situation, as some adopted the city states system of the Southern Hellenes while others remained barbaric in way of life and tongue.
Even proper Illyrians were accepted to the Olympic games and the coins from Skodra (the capital of the strongest Illyrii proprie dicti tribes) were written in Greek, yet it doesn't make them Greeks.
Did we inherit Alexander's biography book where he proudly says 100 times how Greek he was? No.
Did he spread Greek culture? Of course. How different was the Greek culture from it's neighbours? It seems not so much, they're both Indo-European with Balkan influence and right next to each other.
Was this amazing culture we are all fascinated about really Greek (Indo-European)? Maybe not, maybe partially.
Were the true Athenians and Spartans were the same? Maybe not. Maybe the Spartans descended more from pure blooded Dorians/Hellenes and were physically more Northern, while the Athenians were assimilated locals of the Mediterranean type who inherited a way more advanced culture influenced by the even more advanced Aegean, Hittite, Egyptian, Phoenician, etc. cultures.
So Alexander was a cultured rich prince with a hot temper and spirit of a ruthless yet fair and curious conqueror. He wasn't a poor Illyrian ruthless warrior whose main profession was piracy nor a well mannered sophisticated Athenien "softie".
Hope that makes sense and answers the question as we're not talking about modern nations with established borders where even if different, people of the same nation were indirectly "forced" to mix with one another and still create an identity, even genetically. Although in the Balkans luckily that's not yet the case as the nations are still very young.