That's revisionist history on the order of the famous line from "1066 and All That", about how "in 1536, the Pope and his minions left the Church of England". The Vatican was part of a independent state, ruled by the Pope, long before Mussolini was a glimmer in his daddy's eye.
Italy was united as a nation in 1871. Prior to that time, the Pope ruled the central part of the Italian penninsula -- an area much, much larger than the current Vatican City -- as its secular ruler. The nation was called the "Papal States".
After the Pope's armies were defeated by the Italians under Garibaldi, the Pope retreated to the Vatican and refused to recognize the new Italian state. (The term "prisoner of the Vatican" was used to describe the Popes of that era.)
What the concordat with Mussolini in 1929 did was simply to end that irregular situation. The Popes accepted the conquest of most of their traditional territory by the Italians. The Italians accepted the independent existence of Vatican City as a tiny remnant of the Papal States.
Ah, no, thats a fact.