The word Greek in the designator is the oldest thing about it and the most misread. Greek was the working language of the eastern half of the Roman world, the language the ecumenical councils argued in, and the language the Byzantine liturgy was composed in. It marked a civilisation long before it marked a nation, and the churches that carry it were named long before there was a Greek state to be named after.
That is why the largest congregations of two of these churches speak no Greek at all. The patriarchates of Antioch and Jerusalem serve mainly Arabic-speaking Christians of Syria, Lebanon, Jordan, Israel and Palestine, who call themselves Rum Orthodox, from Rum, the Arabic for Roman, meaning the Roman empire that was governed from Constantinople. Their liturgy is celebrated in Arabic. The Patriarchate of Alexandria serves congregations across sub-Saharan Africa and offers worship in local languages. The diaspora archdioceses of western Europe, the Americas and Australia began as immigrant churches and now hold generations who worship in English and in other languages.
Under Ottoman rule the Orthodox population of the empire was administered as a single community headed by the Patriarch of Constantinople. That fixed the association between the Greek hierarchy and Orthodox Christians of every language, and it is where the name the Arabic-speaking churches still use came from. The national autocephalies of the nineteenth and twentieth centuries redrew the administrative map without touching the rite, and the designator outlasted them, because it was never an administrative term in the first place.