Right after I posted the question here, I also put it on a gaming forum I participate in, and one of the ladies there nailed it. Exactly matches a baseball cap rack being sold on Etsy. I was looking at it the wrong way - it mounts vertically, with the slots pointing up, and you slide the strap through the slot, with the caps nesting on each other.
It's a pipe rack now! And I have two of them, so I have room for 20. It was funny when we were in the store and I showed them to my wife, and she kind of said, "are you going to ... fill them?" LOL. "No, Sweetie, but yes."