Q: Why don't Protestants ever claim the Patriarch of Constantinople is the Antichrist even though Catholics and Orthodox share many of the same beliefs?
A: Because the Antichrist has better things to do than listen to the Sultan for 500 years.
The joke refers to how the Orthodox Church (sans Russia) fell under the control of Ottoman Muslims starting with Sultan Memhemd II appointing Gennadios II Scholarios. For centuries the Greek Church was overseen by Muslim rulers.
Today the patriarchs are elected by fellow Orthodox but the Turks still have control. The Patriarch must be a Turkish citizen (due to Turkish law) and a court ruling forbids the Patriarch from using the title "Ecumenical Patriarch" (The Eastern Orthodox Churches recognize the Patriarch as a first among equals). While the Patriarch is trying to reverse the ruling, he seems to be accepting it. How things change how things stay the same.