These rules, if they are rules, seem to bother the press and royal watchers more than the Royals themselves. The King and Queen seem most unstuffy generally. They often greet people they know with a kiss on the cheeks, or, if the King is being particularly gallant, on the hand.
The PM is hardly a stranger to the King and they’ve met many times in the past. I’m sure Sir Kier has been well briefed and if the King were offended by it (which he certainly doesn’t seem to be) a quiet word would have been had. Personally, I think pearl clutching about a hand on the Royal personage, especially when there is no bad intent, seems a little OTT in this day and age.
After all, even that most hands off of monarchs, QEII placed her arm around the First Lady of the United States.