Greek identity is a 19th century creation. People of the East Roman Empire, as well as their descendants in the Ottoman Empire, considered themselves Romans. To them, Greeks (or Hellenes) were the pagans who lived in the region in the distant past. Around the Greek War of Independence, Greek identity started slowly gaining ground. First among the educated elites, partly due to Western philhellenism, and later among the common people.

You can still probably find some older people in rural Greece who consider themselves both Greek and Roman.

As for ancient Rome, it's pretty famous for elevating citizenship over ethnicity. Much in the same way as the US, and unlike most other modern countries. Assimilating conquered peoples and turning them into Romans was a major factor in their success. Other empires that built themselves around a dominant ethnic group were more short-lived. They had to rely excessively on the favored ethnicity for their military needs. Eventually that ethnic group was depleted due to constant wars, and someone else took over.