From my experience LH is very quick to make passengers check baggage if a flight is full, even if not necessary. This probably happens mostly in economy but I could see them asking business passengers to check bags, too. I recently brought two small personal items in short-haul J and was asked to place both under my seat even though theoretically I should be entitled to the space above me in J.

There is no US regulation regarding having only one carry-on bag. LH is just lazy / cheap.