The websites of the Holy See and Vatican are linked below.

Vatican City is not called the Holy See. Vatican City is the home and headquarters for the Holy See - the governance of the Catholic Church. The name derives from the Latin Sancta Sedes - Holy Seat and is the ecclesiastical jurisdiction of the Catholic Church in Rome, the episcopal see of the Bishop of Rome—the Pope.

The Vatican is known as the Holy See

No, Vatican City is not a member of the United Nations. However, the Holy See does have observer status in the UN.