The Byzantine empire was a continuation of the older Roman empire in the East but it was gradually transformed into a different political entity. Meaning:
- The original Roman empire used Latin as an official language, as expected, while Byzantium was Greek-speaking
- They basically inherited the Roman legal system from the Roman empire.
- They considered themselves a Christian state and even the ONLY true Christian state (that's what orthodox means, "one with the right faith").
Having this in mind, the people who considered themselves Byzantines called themselves Romans, which in Greek is "Romaios" (read as Romeos) or "Romios" (read as Romios), which was a more vulgar version, that eventually dominated. In this aspect, "Romiosyni" was used in modern Greek (which is considered the Greek language after 1453) to describe the descendant of the Roman empire, but its main meaning was "all Greek people". For example, in 1945-57 the Greek poet Giannis Ritsos wrote a poem named Romiosyni which has nothing to do with the Byzantine empire (though some might argue that the people it refers to consider themselves descendants of the Byzantines).
In modern Greek though, "Romios" is not used normally and the word used in its place is "Hellinas" meaning Hellene. So, to answer the original question, they gradually stopped using this word (and I mean gradually as the example above is quite recent and it does not suggest an absolute termination) to identify themselves in the late 19th century. I guess the process continued until the 20th century.
The OP asks many secondary questions like
When did the Greek (and Eastern European in general) identity finally
drop its association with the Roman Empire?
This is a different question. The association with Byzantine is quite strong even nowadays in modern Greece and most people consider themselves descendants of Byzantines while there are some that believe there is a continuous connection to the ancient Greek people, Byzantines and modern Greeks.
P.S. The Byzantine empire was a multinational empire (though it cannot be compared with modern nation-states) and there are many people in the Balkans who have some association with its inheritance. The Albanians, for instance, have a flag which is a variation of the Byzantine flag (this does not mean that they consider themselves descendants of the Byzantine empire though).