Politically, I suppose we can categorize the Papacy as an absolute, elected theocratic monarchy, that is, in terms of its governance of the political apparatuses of the Vatican City State and the Holy See.
However, it is difficult to think in these terms in the Pope’s supreme governance of the Church as such. It is not that the Pope doesn’t have absolute authority over the Church in a way analogous to that of a king over his dominion; he does. However, it is a whole different paradigm (cliche, but I think it fits perfectly here). You have to think in totally different terms. The papacy is not inherently and exactly a political institution. It is far more than that and more important than that.
And that is not to say that, depending on the historical circumstances of a given time, the Pope can’t also be an earthly king. He can. Indeed, for most of Church history, Popes have been earthly monarchs. Furthermore, I think that in most cases–although not all–the three hats of Supreme Pontiff, Bishop of Rome, and monarch of the Vatican/Papal States have worked very well together and the latter of these three has, in my opinion, played not at all a negligible role in the propagation of Christianity throughout the world. Temporal authority in the strict governmental sense is not a necessary role for the Pope, and I think you could even argue that it isn’t “proper*” to the Pope, but that doesn’t mean it can’t or shouldn’t be done, nor does it mean it hasn’t produced great fruit for the Church. In fact I argued that it has.
Nevertheless, the three hats, so to speak, are separate. Bishop of Rome and Supreme Pontiff are far more important in an absolute nitty-gritty sense, but monarch of Vatican City State/Papal States is also very important, if only in practical terms.
*Proper as in inherently related. There are plenty of things that aren’t inherently related, but which still are complimentary or work well together for some purpose. This is one of them.
These are my 2 cents.