I can't speak to the Medieval or Renaissance periods, but there were points in which what you call "papal identity" was clearly indicated.

In 1848, a number of revolutions swept through European cities. One of those cities was Rome--seat of the papacy and capital of the Papal States. A Republic was declared, and one of the pope's ministers was assassinated. The pope--Pius IX--fled from the city into exile. Some fled with him, despite the highly uncertain prospects of such a course of action. One of those who fled was Marcantonio Pacelli, who was personally loyal to the pope.

The papacy was restored to Rome two years later (by the French and Austrians, not by the force of arms under the pope) and those who had remained loyal to the pope were rewarded for their service. Pacelli was one of those who was rewarded with a noble title, and this began generations of papal service for the Pacelli family. Marcantonio became a papal minister, and later founded L'Osservatore Romano (the Vatican's newspaper). His son Filippo served as a lawyer for the papacy. Filippo's son Francesco was a Vatican lawyer, and his other son Eugenio was a priest and Vatican diplomat. Oh, and Eugenio later became Pope Pius XII. (The Pacelli family was far from being a peasant family, so perhaps their example isn't perfectly applicable. They weren't alone in demonstrating their loyalty in 1848, though)

Similar divides to those that happened in 1848 were seen in 1860 and 1870. Previous residents of the Papal States (and Catholics in general) had to weigh their loyalty to the pope (and/or the Church) and loyalty to the new regime. A restoration of the papacy wasn't out of the question, especially given the context of 1848, so abandoning one's attachment to the Papal States wasn't a frivolous choice. Loyalty to the papacy lingered well after the Lateran Accords cemented an answer to the Roman Question in 1929, and learning to navigate global politics as a tiny nation took the papacy a number of generations to figure out.

Followup questions are always encouraged, and perhaps another of our aspects can give us some information on the medieval or renaissance periods.