I do not want to belong to any club that would not need to drop a database constraint in order to accept me.

Yes, that's the problem. I understand what he wants to say (basically YAGNI), but imagine the reception desk employee having this problem, escalating it until it reaches a developer just to register John Doe.

I wonder if the parent's reply is actually a joke based on paraphrasing Groucho Marx (the statement itself doesn't have much logical sense).

No he didn't. The "first name/last name" constraint is a straw man for didactic purposes. He's explaining why it never made sense.