Milan's mayor at the time recalls a brief Presidential Suite stop for changing before La Scala. Il Giorno and La Repubblica corroborate the exact property, suite and October 18, 2000 itinerary, but no reviewed source proves an overnight stay.
Queen Elizabeth II used the Presidential Suite at Hotel Principe di Savoia in Milan on October 18, 2000, during an official visit with Prince Philip. The exact property, date and suite are well supported. An overnight stay is not.
Source conclusion:
The cited sources confirm a visit to the hotel, but not an overnight stay.