Before this can be answered, your question should be divided in a couple of sections as there are a couple of misappropriations or possible misconceptions here.

First of all, no one was "Byzantine" during the entire period of what we call the 'Byzantine Empire'. This term was coined in the 16th century to describe the Eastern Roman Empire. The name used by the inhabitants and rulers of this empire stayed The Roman Empire, although in Latin or more often Greek.

The second part that need dissection is the term Africa. After the reconquests under the Eastern Roman Emperor Justinian an exarchate of Africa was set up which ruled until being conquered by the Caliphate. This area never included Egypt, which was part of the Eastern Roman Empire from the first moment the Empire was divided in two. The Exarchate of Africa held out slightly longer than Egypt, which was first conquered by the Persian before being conquered by the Caliphate.

Now to the question of identity. All the African provinces had been part of either a Greek or Roman community for an extremely long time before they were conquered by the Calpihate. The Provinces West of Egypt were conquered during the Punic wars and had been part of the Roman Empire from at least the 2nd century BCE. Egypt was conquered by Alexander the Great in the 4th century BCE and, while not part of a unified empire, was an essential part of the larger Greek speaking world long before it was integrated into the Roman Empire in about 30BCE.

This means that when Egypt and Africa were conquered by the Caliphate most of these Egypt had been part of the Roman Empire for at least 600 years and part of the Greek speaking world for 900 years and the rest of Africa had been part of the Roman Empire for 800 years with a few interruptions in between. Most notably by the Vandals for about 100 years. This means that these provinces were considered Roman for multitudes of time longer than the USA has been a country (242 years). This means that while the local identity was still alive, they all considered themselves part of the Roman Empire. Since inhabitants and rulers of the Eastern Roman Empire did not consider themselves separate from the previous Roman Empire and historians will downplay its difference from the Roman Empire, these provinces considered themselves largely "Byzantine" as you put it. Although I do hope that you now understand why that term is problematic.