Victor Emmanuel III (1869 -1947) was actually the third - and second to last - King of Italy, who had little to do with the creation of the flag... something older than he was. For what concerns the reason as to why the Italian one sported the House of Savoy's own coat of arms, the simplest answer is that Piedmont (the state that ended up uniting the entirety of the Peninsula) already had that symbol on its flag as it was ruled by that very same dynasty.

Why wasn't another state's emblem chosen? Well... it was not them who united Italy in the first place!