Why does the Greek Orthodox Church have religious hegemony in Greece?

Start with Byzantium: Orthodox Christianity was the state religion, and heterodoxy was deemed treason. Jews and Muslims were tolerated in Byzantine Law as second class citizens; heretical Christians got the sword.

In the Ottoman Empire, that continued with the Rum millet: Greek Orthodoxy defined the nation of Romans, which was considered to include Greeks. Catholicism was a minority presence in Greece, and Greek Catholics were deemed not Rum (Romioi, Romans), but Frenk (Frangi, Franks).

When the Modern Greek State was founded, Orthodoxy became the state religion quickly; and it was considered coextensive with Greek national identity. That has allowed it a hegemony that Western Europeans are uncomfortable with; the Church of Greece gets veto, for example, on building places of worship for any creed, which is why there still isn’t a mosque in Athens. Is the 180 Year Wait for an Official Mosque in Athens Finally Over?

Catholics were ignored, and they were small enough in numbers that they could be ignored. Muslims were Turks as far as everyone was concerned, whatever their ethnicity (Turkish, Gypsy, Greek, or Albanian). Armenians were foreigners. There was some Protestant missionary activity in Greece; the Ottomans considered them a distinct millet, and the Greeks… well, the Greeks ignored them too, just like they ignore Jehovah’s Witnesses.

So, partly history, partly construction of national identity, partly privileged role of the state religion.