I don't think they are "gaffes" at all, I think he's just very funny, with a wicked sense of humour. I was watching some old footage on YouTube of Groucho on live TV game shows recently (What's my line?) and he said similar things and everyone thought him hilarious - which he was.
I know repartee is defined as what you wish you'd said, but wouldn't it have been wonderful if the lady had replied - The State and the British public, the same as you and your extended family!
I am sure His Highness would have been holding his sides!!!