In addition to what has been said, Italy has sworn to protect the Papal states from invaders, so they are a state that has even fewer attributes that one would ascribe to a sovereign state.
Bear in mind that the Holy See was EXTREMELY political in the earliest days, and has gradually lost direct jurisdiction over former “Catholic lands”. The organization remains politically involved in a nuanced way, but it has decided to market itself in a more spiritual light than its former political one (they didn’t even have that many doctrines until the Reformation convinced them to specify out of spite).
They found out how to remain influential without burning people at the stake, and that is by “spiritually” persuading millions of grandmas to fork over a segment of their fixed income, and pressuring their kids to propagate as much as possible and to baptize their young, and put the Catholic yoke on their neck with a spiritual fear in all of them, never giving them assurance of salvation. This, they hope, will make Rome fat and jolly with additional wealth, and it does.
Because it is such a wealthy entity and due to the fact that Italy basically functions as its military, because thousands of people dangle rosaries in their hands and are ready to do the bidding of Rome, and the fact that it [arbitrated between Argentina and Chile in a border dispute](https://www.nytimes.com/1978/12/24/archives/argentina-and-chile-accept-papal-effort-in-dispute-45000-chilean.html), I would say they are transcendent of states since they have a monopoly on certain minds as opposed to direct political hierarchies of foreign states.
You ask “why the Vatican is independent?”, but I ask “who is independent of the Vatican?”
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