So I just noticed that this thread has gained some legs since I first asked the question almost nine years ago. I’ve skimmed through the last two or three pages, and what I take from it is this: There are two answers: the legal one and the practical one that applies in real life.
The legal answer is that there is no legal authority.
The practical answer is that if you question it or mess with them they’ll make your life difficult. On this point, it is well known that one must beware of big power in the hands of small people. This problem is particularly acute with customs and immigration people, IME.
Since misbehaviour among customs people is prevalent, the reason for the “policy” is probably that they don’t want you recording their faces and actions. Same reason many airline employees remove or cover their name tags.