I am currently studying medieval Europe and was always confused by this seeming contradiction. Historically, many Popes defined themselves as spiritual leaders, and although they often did mettle in European politics, they tended to project to other kingdoms that the Church would not interfere with secular matters.
And yet, the Pope himself was the monarch of a large portion of central Italy: the Papal States. I understand that this rule was challenged by many kings and clerics, but how did Popes justify to the rest of Europe that they were allowed to rule their own kingdom?
Although it's not exactly what you're asking about, maybe this thread's comments by u/sunagainstgold can help clear up some of your doubts. In short, ideas and understanding regarding sovereignty were fundamentally different in the past; although it might seem logical to us, in the past there wasn't always a need or desire to separate religious and secular authorities.
Although the Catholic Church and the Papacy were institutions that impacted the whole of Western Europe, many of the Papacy's temporal quirks were in fact a consequence of the unique the political system in Italy: most of the Italian states people are familiar with (like the Grand Duchy of Tuscany, the Duchy of Milan, and of course the Papal States) have their origins in the 10th and 11th centuries, when collaborative sort of government emerged uniting landowners, urban professionals, and the local clergy in authority structures located in urban spaces. I actually wrote about the Papal States' development compared to other Italian states here in a somewhat rambling answer.