The "idea of Rome" is not tied to blood or religion. The Roman Empire and the Eastern Roman Empire had more than 40 dynasties, many non-dynastic emperors, military emperors, et cetera. In regards to religion, Rome had Roman Polytheism, Greek Polytheism, Greco-Roman Polytheism, Egyptian, Syrian, and Oriental Cults, Gallic and Iberian Cults, then with Christianity a multitude of doctrines, Chalcedonian Christianity, Arianism, Monothelitism, Iconoclasm, Orthodoxy, and Uniate Orthodoxy/Catholicism. Mind that Constantine Palaiologos subordinated the Orthodox Church under the Catholic Pope, which made him extremely unpopular. Funnily, Orthodox Greeks were celebrating the arrival of the Ottomans, since the Ottomans restored the historic independence of the Patriarchate of Constantinople. So what is Rome? It is a pluralistic, ecumenical state, surrounding the Mediterranean, organised around a massive and thorough imperial bureaucracy and a professional or semi-professional military force to defend and expand its borders. The only state which managed to achieve this following the decay of the Roman Empire after Michael VIII was the Ottoman Empire. Now in regards to blood, we should mention that the Ottoman dynasty carried the blood of several Byzantine princesses. In regards to titles, the Ottoman Padishah was referred to as the "Caesar of Romans" too, a title which was given to the Ottoman dynasty by the Ecumenical Patriarch of Constantinople, following the fall of the city.
I apologise for derailing the thread, but Orientalism is annoying.