I thought it was equal to looking like Singapore

Obvioysly, but the point is that is's not only a unique circumstance, it's not at all the norm for "Christendom." The Pontifex Maximus is the ecclesiastical authority, the Emperor the secular one, or with time all the various kinds and princes of different realms which collectively made up Christendom. Even the rule over the Papal State became something tied to the separation of church and state by becoming a power base for the Pope where he could assert his independence from the Emperor or secular rulers. Rome could thus also become a neutral ground for the international system. While many kings and emperors would try to bring the church/pope under their authority and occasionally they even succeeded in doing so, in the long-run it became practical for everyone to support and protect the independence of the Pope just so their rivals wouldn't have control over the church. In any case even in medieval kingdoms, the church owned land and could thus have serfs, or certain territories might have been governed by a "prince-bishop" so it's not like the church having some secular authority was that strange. The separation of church and state, while it existed in the middle ages, was a lot fuzzier than the clean-cut idea we have of it now, and needs to be understood in a context where secular states still had and generally enforced an official religion.