I don't see many instances of this being asked so ill say it's not a troll question. The only answer ive really seen is just that Greece was "different". I don't see how it was more different than say Egypt though, in fact all of the cultures in the ANE were pretty distinct.
What seems to be the point of the term is how they built off each other culturally and technologically, and i don't see how Greece is exempt. I hadn't given it much thought until i heard that part of ancient Greece, the Minoans, are grouped in but not Greece overall. Why are they exempt despite having a Semitic alphabet, worshiping in many cases the same gods as their neighbors, and even including or assimilating them into their important myths to the point of naming their part of the known world after a mythical Phoenician princess?
Geographically speaking, Greece is not included in either 'Near East' or 'Middle East', because it is considered to be situated in the 'Middle'. The ancient Greeks had a myth, that the middle / center of the world was in Greece, and specifically at Delphi. According to the myth, Zeus left 2 eagles to fly from the 2 ends of the world (West and East) and they met over Delphi, signifying that there was the middle of the world. The myth was taken by Romans, and the Romans spread it all around their empire, and so even the sea Greece lies in was named 'Mediterrenean' meaning 'Middle-Earth' because of that view.
Because of the way historiography has played out. The modern academic tradition for the most part descends from Western European academic tradition which derives ultimately from the Greco-Roman intellectual tradition. Thus we inherit their own conception in which Greece is defined differently from the Mesopotamians and Levantines, which of course links to the whole distinction between the Orient and Occident.
Tl;dr- It's an artifact of historical bias
There are some good answers here already, but I'll just add that there's some really interesting scholarly debate about this very subject. Were the Greeks really obsessed with defining themselves as different from 'barbarians' in the east (aka Edith Hall's Inventing the Barbarian or Paul Cartledge's The Greeks) or was it a later post-enlightenment convention to firmly establish the Greeks as fathers of Western civilisation (as per Martin Bernal's Black Athena)?
These questions have fuelled some fascinating debates on the historiography of the classical world. Personally I cowardishly and pragmatically come down in the middle - I think it's clear that the Greeks really defined themselves through their difference to Eastern neighbours, although in reality they were closely related. I think it's also been useful/easy for the modern west to maintain an 'East and West are fundamentally different' narrative in it's telling of the past.
For more, you could do worse than explore the heated and passionate discussions in Black Athena, Black Athena Revisited (against the thesis) and Black Athena Writes Back.