Pope Francis was born to Italian parents -- who ran from Italy when the Axis powers unified. His father's Italian name is the equivalent of the English Peter ... no immediate record of his being from Rome ... but that connection, in terms of you are who your parents were (as applied to Jewish at the time Francis I was born) might be applied.
The real key is Rome -- it was known as The City of the Seven Hills, and is the only city in antiquity to which that designation was common ... especially in Europe and therefore for Malachy to use. Will the Vatican be destroyed? That is the destruction of Rome in terms of the Rome as it existed in Malachy's age. That would mark the end of the RC Church... at least effectively... especially if the cause was an earthquake whose destruction was complicated by flooding associated with global warming, or a relatively permanent population shift ... the relocation of the Papal Throne to another nation or city.