A lot of responders have been making this point. It is true that when I am creating an NPC for the campaign, I don't keep a close watch on encumbrance limits. (I think most DMs would agree that we've got better things to spend our planning time on than counting up every half pound of equipment.) And it's true that I don't hound the players, force them to turn over their character sheets, and verify that they are adding things up correctly and taking appropriate penalties. Again, I've got better things to do. But if you are going to follow a rule (as my players are apparently willing to do), then you should follow it correctly and not try to milk it for every ounce of advantage you can get.
Here's a question for all the people who have sided with my player: Suppose the character in question had found some bracers instead of a cloak. Bracers weigh 1 lb. Could my player have said that he was throwing away the cloak from the explorer's outfit (which apparently weighs a pound) to leave his weight limit unaffected? If not, why not? They are both worn. What if it were a flask of oil (which also weighs a pound)? Sure, it's not clothing, but it will be kept in his vest pocket (assuming he hasn't already thrown his vest away to trade it for some other more useful item).
I think you see where I'm going here. Trying to say that
I'm being unrealistic by refusing to let my player trade cloaks and boots for free is backwards. If anything is unrealistic here, it's the rule that says your outfit weighs nothing. (And let's remember that the rule is very specific. It doesn't say that whatever you wear weighs nothing. It says, "The first outfit is free of cost and does not count against the amount of weight a character can carry." Only the starting outfit has this property, not anything you choose to wear.)
Now just because a rule is unrealistic, that doesn't mean it's a bad rule. As I said before, I think this rule makes perfect sense from a game mechanics perspective. Since the initial outfit conveys no game benefits, it should not convey a game penalty. (Otherwise, everyone would run around in a monk's outfit all the time.) Those people who try to justify this rule in terms of some version of "realism" are just plain wrong, IMHO.
Anyway, I started this thread to get some feedback, and I got it. Thanks to all of you who cared enough to respond. Special thanks to Diaglo, though, for putting the whole thing in perspective.

Diaglo, I hope your character only carries a
countable infinity of slings around with him. Otherwise that infinity times zero equals zero stuff may not work any more.