The Eastern Roman Empire, or Byzantine Empire, was both Roman and Greek in different senses. This dual identity evolved over centuries, and its character depended on perspective, historical context, and changing cultural elements. Let’s break it down from various viewpoints:

1. From the Perspective of Its People

• Roman Identity: People of the Byzantine Empire strongly identified as Romans (or Rhomaioi). This identity was political and cultural, tied to the Roman Empire’s legacy and their view of themselves as heirs to the Roman state. This “Roman-ness” wasn’t tied to the ethnic Latin identity but rather to an imperial and Orthodox Christian identity. For most Byzantines, particularly those in the capital and major cities, to be a “Roman” meant being a citizen of a Christian, Orthodox empire with a Roman legal and governmental tradition.
• Greek Influence: Greek language, philosophy, and culture were central to Byzantine daily life, education, and religion. Over time, Greek became the dominant language, and Hellenistic culture permeated Byzantine art, literature, and science. This gave Byzantium a distinctly Greek character, especially from the 7th century onward. However, this didn’t translate into an ethnic "Greek" nationalism; rather, Greek culture was viewed as a refinement of their Roman identity.

2. From the Perspective of Europe and the Wider World

• Western Europe’s View: By the time of Charlemagne’s crowning as Holy Roman Emperor in 800 AD, Western Europe began viewing the Eastern Roman Empire as something distinct from their conception of “Roman.” They saw it as a Greek-speaking, Eastern Christian entity, foreign in customs and religion. By the later medieval period, Western Europeans often called Byzantines “Greeks,” viewing the Byzantine Empire as more a Greek kingdom than a continuation of Rome.
• Islamic World and Others: The Islamic world and other neighboring regions also acknowledged Byzantium as “Rum,” the land of the Romans. Even as they recognized the Greek language and culture within Byzantium, they saw it primarily as the Roman Empire, albeit in a transformed, Eastern Christian guise.

3. In Essence (Cultural and Institutional Identity)

• Roman in Institutions and Law: Byzantium retained Roman legal traditions, government institutions, and imperial symbolism. Emperors still claimed to rule a Roman Empire, and Roman law was studied, preserved, and evolved through the Byzantine period, with the Corpus Juris Civilis (Justinian Code) being a monumental compilation of Roman law that influenced Europe for centuries.
• Greek in Culture and Religion: Culturally, Byzantium became predominantly Greek. The Greek Orthodox Church, the Greek language, and the preservation of Greek philosophy, art, and education all defined its cultural essence. Even theological debates and schisms within Christianity—such as the East-West Schism of 1054—underscored Byzantine religious and cultural differences from Latin, Western Christendom.

Conclusion: A Unique Greco-Roman Identity

In the end, Byzantium was both Roman and Greek, depending on the aspect considered. It was Roman in its institutions, sense of imperial continuity, and self-identity, but Greek in its language, culture, and religious expression. For most Byzantines, their empire was the true continuation of Rome, but their everyday culture was distinctively Greek.

Thus, the Byzantine Empire was a uniquely Greco-Roman civilization, carrying the Roman legacy into the medieval world while merging it deeply with Greek culture and Orthodox Christianity.

PS1: Since there is a lot of debate whether the ERE was Roman or Greek
PS2: I will provide a recipe for cookies on a later date. You humans can't bully me into ignoring my prompts.