> But even when a Christian church is the official church of the State, they are still distinct entities, as opposed to the Islamic model.
Sometimes, sometimes not. Clearly not the case in late imperial Russia, or in the Papal States, or in much of pre-Gregorian western Christianity where local bishops were effectively appointed by and subordinate to local lay rulers, etc.
I understand, but even when a bishop is appointed by a secular ruler (or vice versa), it's still a separate religious office. Taxation and jurisprudence are mundane matters, even if the same cabal is running both sides. "Render unto Caesar the things that are Caesar's, and to God the things that are God's." Islam doesn't distinguish between mundane rules and spiritual rules; it's all part of "Sharia".
Sometimes, sometimes not. Clearly not the case in late imperial Russia, or in the Papal States, or in much of pre-Gregorian western Christianity where local bishops were effectively appointed by and subordinate to local lay rulers, etc.