You mean the Greek subjects of an emperor who spent his time adventuring past the known extent of the world, with all the communications technology the world could offer 2400 years ago? The Greeks who split off from the Asian part of Alexander's empire immediately after he died in a foreign country thousands of miles away?
The empire split after Alexander's death into the Macedonians proper, the Seleucids in Asia, and the Ptolemies in Egypt. The Ptolemies certainly kept a lot of cultural continuity with the Greek / Macedonian world -- so much so that the last Ptolemaic pharaoh, Cleopatra, is believed to have been the first one who was able to speak the Egyptian language (after ruling Egypt for 250 years!). They did set up syncretistic temples, which was standard practice basically everywhere in the classical period. That was an effort to get the Egyptians to accept Greek religion, not the other way around. The Ptolemies even set up a major new city for Greeks. Their accommodations to Egypt weren't particularly impressive.
To the extent that the Greeks and the Macedonians had a common cultural identity, it was because the Macedonians aspired to Greek culture (it seems safe to say that the Ptolemies eventually succeeded at this). The Greeks didn't think highly of the barbarians to their north.
The Romans, IIRC, were uncommonly willing to allow subjects to participate in Roman society as long as they followed the rules. This did not mean that the Romans themselves changed their own lifestyles.