An AAAA cell has 200-350 mohms internal resistance. A 9v battery has 6 of them in series (many of them are literally that, others have their cells as a stack of plastic buckets). The nose ring is a short run of wire, it’s idunno a 0.2 ohm heater?
Dry skin is hundreds of thousands of ohms. Even wet skin has pretty good resistance. When you touch a 9v to your tongue, you’re starting to mess around with lower resistance flesh, it is definitely not a comfortable thing to do.
The metal ring doesn’t do anything to move more electricity into your body, but it soaks every electron the battery can push and turns it into heat. Best I can figure it would amount to a few watts, which would be toasty if you were holding it between your fingers. The septum is a thin piece of flesh, I think it would sauté pretty quickly.