What made Byzantium "Roman" other than its formerly being part of the Roman Empire? How Roman was it when it spoke Greek and, after the 11th century, was no longer Catholic?

I've never understood that either! Geographically and linguistically, they were much closer to the Greeks.

Byzantium, later known as the Byzantine Empire, was indeed a continuation of the Roman Empire in the East. Here's what made Byzantium "Roman" despite its differences from the Western Roman Empire:

1. Roman Legal System: The Byzantine Empire adopted and continued to enforce the Roman legal system. The "Corpus Juris Civilis," also known as the Justinian Code, was a comprehensive legal code compiled during the reign of Emperor Justinian I. It served as the basis for legal principles throughout the Byzantine Empire.

2. Imperial Administration: Byzantium maintained a centralized imperial administration similar to that of the Roman Empire. It had an emperor with an elaborate bureaucracy governing the state and its territories.

3. Roman Heritage: Byzantium consciously sought to align itself with the Roman legacy. Emperors styled themselves as "Augustus" or "Caesar," claiming the imperial title used in Rome. They considered themselves the rightful heirs and protectors of Roman traditions and culture.

4. Roman Architecture: Byzantine architecture exhibits distinct Roman influences. Structures like the Hagia Sophia in Constantinople (now Istanbul) demonstrate a continuation of Roman architectural techniques, particularly in terms of domes and vaulting.

5. Continuation of Roman Institutions: Byzantium carried forward many administrative and governmental systems of the Roman Empire, such as the Senate and the imperial court. These institutions, while evolving over time, still retained Roman characteristics.

It is true that Byzantium gradually adopted the Greek language and became predominantly Greek-speaking, and there were significant religious differences between the Roman Catholic Church in the West and the Byzantine Empire in the East. However, it is essential to note that "Roman" did not solely refer to ethnicity or language but encompassed the continuation of Roman legal, political, and cultural traditions. Despite linguistic and religious changes, Byzantium maintained a considerable degree of Roman influence and identification.