There was no "Vatican" back then, I mean nobody used this term to mention the pope. In 1788 the Italian peninsula was divided into several states, like the Republic of Venice, the Kingdom of Naples, etc... and the Papal States in central Italy. The Papal States were a sovereign country with its own institutions, its laws, its army, its fleet... Your coin is a coin of the Papal States. In the 19th c. all Italy was unified and became the Kingdom of italy, with Rome for capital. The pope did not accept this, but he was toppled as a political head of state and took refuge in the Saint Peter Basilica. He was now a spiritual authority, but no longer a political ruler. The situation was settled under Mussolini when the pope signed the Laterano Agreements with the Kingdom of Italy : a "City of Vatican" was created in an area of Rome, consisting of the St Peter Basilica and the nearby blocks, and the pope is its political head of state. This City of Vatican is internationally recognized as an independent city-state, one of the world's smallest in terms of surface but with a diplomatic network all around the world like a superpower.
