Why does a permanent, non-moving magnet, stuck to a long piece of metal generate a voltage?

You are not actually measuring anything about the circuit because the leads are plugged in incorrectly.

The negative black lead is stuck into multimeter current measurement input, and positive red lead is stuck into multimeter voltage measurement input. The multimeter common ground is floating, so the multimeter is not able to actually measure voltage with the meter leads.

It is just the multimeter picking up some fluctuations because it's common terminal is not connected to anything. It might show similar readings with the meter leads unplugged.

My guess is it's a Galvanic cell. One electrode is your metal strip (maybe nickel), the other is your magnet whatever it's made of (or coated). The electrolyte is probably sweat or saliva (you used your tongue as a probe). When your electrolyte dried up, voltage went down.