That white cross on red is the coat of arms of the House of Savoy, to which the kings of Italy belonged. This is a medieval noble family who originated as counts of Savoy (hence the name), a region that nowadays is split up between France, Switzerland and Italy.

Where they got the cross from is uncertain but is postulated to originate from the Crusades, where many crusaders used similar flags. There is no known relationship with Denmark, though the origin of the Danish flag is uncertain and may also come from the crusades. Interestingly, the stretched "Nordic cross" version of the Danish flag that is in use nowadays is an early modern innovation, and in late medieval times they would have used a more conventional cross almost indistinguishable from that of Savoy.