Why can't a multimeter measure its own current and voltage?
It's because of the circuit used inside of the multimeter to create a virtual ground at approximately Vbat - 6.2V, or about 2.8V for a 9V battery; this basically means the COM terminal on the meter is at 2.8V relative to battery ground and readings will be offset by this.

I tried with another meter to measure its 9V battery - it read 6.08V, which is very close to the predicted 2.8V drop.