This truly is an interesting issue if we consider "identity" through the faith (Orthodoxy)
Actually in the ME and the Balkans, under the influence of the Ottoman system of Milliyet, national identity or ethnicity was and if I may say is defined by faith; People were bound to their milliyets by their religious affiliations, rather than their ethnic origins. So, in the population exchange after the 1922 catastrophe of Minor Asia, and according to the Lausanne treaty (1923), you have Muslim Cretans, ethnically Greeks, perceived as Turks and therefore moved to Turkey, while on the other hand, Christian Cappadocians, a large persentage of whom did not speak a single word in Greek, perceived as Greeks and moved to Greece. Ethnicity in the Balkans, is by largely defined by faith. The language comes second. One of our heroes in the revolution of 1821, Athanassios the Deacon, after the unlucky for the Greeks battle of Alamana bridge (near Thermopylae), was captured and brought before the Turkish Pasha alive. According to tradition, the conversation they had, goes like this:"Turkish Pasha: Will you become a Turk, deacon? (note, the Pasha asks him not to become a muslim but a Turk).
Athanassios the Deacon: I was born a Greek, I shall die a Greek"
(the rest of the story goes like this: Athanassios was impaled and roasted alive οn a spit (literally BBQ-ed). At the time of his sacrifice he was 33 years old).
In the period of the Byzantine Empire, Greeks referred to themselves as "Romans" although they were Greek. But the liturgical practice was conducted in the Greek (at least in the East). Did these Greeks have a loss of identity? I don't believe so otherwise we would not be calling ourselves Greek today.
Yes, in the Byzantine Empire, Greeks did indeed called themselves "Romans"; but "Romanity" had nothing to do with ethnicity. It was mostly a cultural (Orthodox Christians who spoke Greek) and not an ethnic identity. Besides until 11th-12th century AD, the adjective "Greek" was used in reference to the Heathen Greeks. However, one must stress that Byzantine Greeks, felt the continuity they had with the ancient Greeks; they felt it & wrote down that the historic link between medieval Greeks and the ancients had never been broken:
writes in her work "The Alexiad"
(after she's boasting about how well she speaks & writes Greek; she's proud she's not using her contemporary colloquial dialect but rather classical Greek in her writings, "the language of Aristotle" as she says. She describes the Byzantines as Greeks (Hellenes): "There could be seen a Latin being trained, and a Scythian [she probably means a Slav] studying Greek, and a Roman handling Greek texts and an illiterate Greek speaking Greek correctly"
(Book 15:7-9) (in Greek: "Καὶ ἔστιν ἰδεῖν καὶ Λατῖνον ἐνταῦθα παιδοτριβούμενον καὶ Σκύθην ἑλληνίζοντα καὶ Ρωμαίων τὰ τῶν Ἑλλήνων συγγράμματα μεταχειριζόμενον καὶ τὸν ἀγράμματον Ἕλληνα
wrote this to Emperor Manuel Palaeologus, for the people under his reign: "We are Hellenes (Greeks), as the language and the ancestral education testifies"
(in Greek: "Ἕλληνες
ἐσμέν τὸ γένος, ὧς τε ἡ φωνή καὶ ἡ πάτριος παιδεία μαρτυρεῖ).
-Byzantine historian Laonicus Chalcocondyles
in his work "Demonstrations of History-Historianum Demostrationes"
(in Greek: "Ἀποδείξεις Ἱστοριῶν"), strongly believed in the connection between the ancient Greek civilization & his contemporary one. He describes the people of Byzantium as "Hellenes"
Returning to the subject of identity, can we really say we will lose ourselves as "Greeks" (or anything) as long as we have Orthodoxy?
Not only Orthodoxy, but the language also is what defines us as Greeks:"GREEK, the language they gave me; poor the house on Homer's shores. My only care my language on Homer's shores..."Odysseus Elytis, 1979 Nobel laureate for literature