No, there's nothing in the USB spec that requires the internal processor to be updatable. The headline is clickbait.
To get this to work, you need to solder wires onto the USB flash drive's PCB, and then program a "burner" firmware image into the microcontroller. Only then you can remove the wires, and further program its firmware over USB. The stock firmware for nearly any USB device would never allow itself to be upgraded over USB.
This strategy allows you to give someone a malicious flash drive that enters keyboard commands into the computer (this has been done before), or infects the computer by pretending to be a device whose driver has a security flaw. It would not allow you to infect the other USB devices connected to that computer, because they don't have burner firmware.
Yes, you're right - I've done some more research on the github site, and figured out how they do it. The burner image can be sent to RAM over USB, to cause the microcontroller to boot from it. Then, that burner is used to flash the malicous firmware image.
To get this to work, you need to solder wires onto the USB flash drive's PCB, and then program a "burner" firmware image into the microcontroller. Only then you can remove the wires, and further program its firmware over USB. The stock firmware for nearly any USB device would never allow itself to be upgraded over USB.
This strategy allows you to give someone a malicious flash drive that enters keyboard commands into the computer (this has been done before), or infects the computer by pretending to be a device whose driver has a security flaw. It would not allow you to infect the other USB devices connected to that computer, because they don't have burner firmware.