Cyril and Methodius were Roman citizens, as was their father. Whatever ethnic roots they had was a secondary concern, much in the same way as it is in the US today.
People always try to think about the past in today's terms. No, it was's anything like today's America and Roman citizenship wasn't anything like today's citizienship of a country either. Ethnicity did matter more back then than it does today. Tell me, what did "Greek" mean back then and did it mean anything and were they Greeks? If it's the citizenship that matters, why so many Roman citizens presented as "Greeks" today?
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.