I checked out the location of Residenza Paolo VI Hotel on several sites, including the site of the hotel. Nowhere it is claimed to be in Vatican city, it is however claimed to be on St Peters Square.
According to Google maps that is right, just outside the Vatican city borders.
Here is a link to the map.
More searching on booking.com does not result is hotels within Vatican city.
They place a marker on the middle of St Peter Square but when you follow the marker it is for an apartment rental site and non of the offered apartments are very close to the square, even less within Vatican city.
If there is a hotel within the boundaries of the Vatican City, I bet it is a private (church owned) one for visiting officials, not available for commercial trade. But as I only work from internet, I am not sure.