NiMH discharge protection and MCU powering
The dropout voltage of 1V is arbitrary. The datasheet says it is 1.1 to 1.3V at 0.8A load, and the dropout is measured when Vout has already dropped by 1%. Dropout voltage is larger if current is larger and smaller if current is smaller.
The voltage regulator does not stop operation at the dropout point, it just can't keep the output at the rated voltage when there is not enough input voltage.
So no, it is not enough to prevent batteries from draining completely. That regulator has no undervoltage lockout.