I've decided to post more detail here in case others have the same issue. My dealer is still balking at the repair because Mazda apparently doesn't recognize it as a problem.
First of all, the USB stick I'm using is PNY 64GB Classic Attache USB 2.0 Flash Drive Model P-FD64GCLCAP-GES3 that I picked up from NewEgg last August. I loaded it with MP3s (and probably some cover art and play lists, too), and started using it in the car. Other than the few minutes it takes to read and catalog the music on the USB stick, it's been working just fine in the car from last August until earlier this month. It's been plugged into the car exclusively (never used in any other car or device, and no changes were made to the content) since August and until it started reporting "USB Error". After that, I took it out and tried it in the computer and it appeared to work fine.
I called Mazda "Customer Experience Center" (800-222-5500) for advice on what to do next. They had a record of the reported problem and noted that my dealership even called in for help. Apparently the dealership was supposed to tell me to call the "Bluetooth Hotline" to open a case there. This is because a different company (I think it was called "Mazda Handsfree") handles repairs for the bluetooth-related issues. I asked if the dealer had already called the Bluetooth Hotline, and the customer service guy explained that it's something the owner has to do.
I called the Bluetooth Hotline at 800-430-0153 and explained the problem again. They said that USB issues are usually due to incompatibility, but agreed that this was clearly not the case here -- because it worked fine before the car started saying it didn't. He said he opened a case, and that I should bring the car back to the dealer and have them call the Bluetooth Hotline, using my phone number as a reference number. He said that they would help them go through a diagnostic routine that starts with them swapping out the USB port and goes on from there. He assured me that they'd get it fixed as soon as they figured out what was actually causing the problem.
So that's where I am now -- I'll bring the car back next week and see what happens...