Drunkard’s cloak. Pillory.
OK, so how many people (SD members honorably among them) know that “cooperage” is also a word?
Or you could use bags. [Video link]
Whatever happened to that much superior column and website mentioned earlier in this thread?
I noticed that. 
The topic of the origins I can’t add anything to, but the idea is that the sap in question “lost his shirt”; i.e., had to sell his clothing (or turn it over as part of his debt). The trope obviously made much more sense back when clothing was expensive and even secondhand clothing was worth selling. I know a bit about clothing from the turn of the (last) century, and I’m pretty sure that around then you could buy a used barrel for less than the price of a secondhand suit.
Nitpick: those are proper ladies’ bathing suits of the era, not underwear. The film is supposed to be slightly salacious, as many of the early films were, but at a level suitable for general consumption, which a film of ladies bathing in their underwear would not have been.