Byzantine was Western Christendom even when Constantine, Roman Emperor, moved the capital of the Roman Empire to Constantinople because of the tyranny exercised by the Praetorians over Roman emperors as a rule. This developed over the time since the demise of Rome as a Republic, where emperors weakened the Senate and established the Praetorian Guard to maintain their power over any Senate that remained. Constantine was interested in putting the Praetorians back in their proper place, and they were having none of that.
In terms of religion, every Roman leader whether Caesar or Augustus by title, was anointed by whatever head god there was, and that included other head gods besides Jupiter. Every time Rome conquered new lands, the Romans also adopted the religions of the people they conquered and incorporated them into the official state religion, which is why there's "greco-roman" mythology with corresponding gods. This wasn't always the case--it was the case only after Rome conquered Greece. Same was true when Rome adopted Christianity as official state religion and made adaptations to it as well.
This is also the reason why there were Ecumenical Councils at all, presided over by Constantine initially, and why these councils determined on an official basis which type of Christianity were accepted by the state or deemed to be heretical...case in point, the followers of Arius, which came up as a specific issue at more than one Ecumenical Council.
Different versions of Christianity were still being disputed when Mohammed was born, and Mohammed was something of a fundamentalist who favored original Judaism, as in lineage traceable to Adam, then Abraham, which isn't unusual considering what a big deal Christians made of Christ's lineage to King David despite claims of having lineage no further back than God himself, directly. Mohammed spent a lot of time praying and fasting in the mountains of the area when he found his direction, but found it important to say that he himself was not divine or had any other divine connections besides the Angel Gabriel.
So how did Islamic government happen? It was still tribal in nature; Islam did not usurp existing tribal government, and it's for that reason that a lot of Romans, under imperial divine-established tyranny, found Islam preferable. It is not true that Islam spread "by the sword". It was Sharia Law that reconciled tribal practice with Islam, whether it be Sunni or Shiite version of Islam, and that's how government occurred in Islam; the thing that united all the tribes into an Islamic Empire was trade across north Africa & south down its coasts, from Spain to China--and the sciences. Tributes were paid up the chain of command, as it were, to the high officials in the capital cities, but government was primarily local.
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