Italy originally wasn't controlled by one ruler. It was split up into a bunch of little ministates, each of which had their own main town. So Venice (and the surrounding area) would have its own ruler, independent of Rome's ruler. Or at least I think that's how it was. Now there are only two ministates in Italy: San Marino and the Vatican.
Just as clarification- I live in Italy, and it can be quite confusing whensomeone asks "from which Paese are you?" ! It's more literal translation, and how think of it, is land. So that is how it refers to any sort of land/region (states, counties) seemingly without changing meaning.
Not all cities in the UK have a cathedral by any means, particularly the newer ones, eg Brighton, Wolverhampton and Preston (which last does have a cathedral though not of the established Anglican Church - makes a difference). They were awarded city status by virtue of their size.