According to the story on one small island, the local children asked who the soldiers were. They said "we're Greek, like you!" The children were confused and answered "no, we're Romans!".
To be pedantic, the soldiers might have probably said Hellenes, not Greek (γραικοί). But Roman was also a common ethnonym back then, used by the Ottoman society. In Greek (Helleneke) it was spelled Ρωμιός https://en.wiktionary.org/wiki/%CE%A1%CF%89%CE%BC%CE%B9%CF%8... . The point of confusion was probably that, I suppose?
I grew up in Greece, no one uses any name other than Hellene anymore. To the point that "Greek" sounds more English than Greek to a native Greek speaker.
"Romios" or "Romiosyni" (the act of being a Romios) is considered in many instances parallel to being Greek.
Remember that the Roman Empire dumped Latin and went with the Greek language.