It certainly is true that the French carry out these checks. I was asked for all my aircraft papers at Carcassonne last month while flying a friend's N-reg SR20.
Fortunately I had everything they wanted with me, since I don't think I'd have been popular if I'd returned sans aircraft!
Actually I was slightly surprised it only happened once during the week I was in France, given the recent friction between that country and the US. I had been expecting the 'N' on the fuselage to attract more unwelcome attention than that.
It has never happened to me in a G-reg aircraft but I always take the full stack of paperwork just in case.