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The pope is generally not called the Holy See, he is the bishop of the Holy See. The pope lives in the Papal Apartments located in Vatican City, and has a summer residence in Castel Gandolfo.
The Holy See is a sovereign entity, but is not a nation: it is a diocese that has universal jurisdiction. The territory of any diocese aligns with national/political borders only for practical reasons. If a church building were able to separate its church grounds from its country, it might become a sovereign state, but it would still be a part of its diocese, under the authority of its bishop.
When someone says "the Vatican" they might be talking about either the Holy See or Vatican City. Note the Catholic Encyclopedia article on the Holy See, part of which says:
[...] We can thus readily understand how Holy See came be the technical term for the pope, the central ecclesiastical government, and the actual abode of the same.
It is probably best not to copy this potentially confusing practice (see synecdoche).
For more information see the section on jurisdiction in the Hierarchy article, and the article on the Pope.