To quickly understand bridging, two amplifiers are used and 1 is fed the non-inverted input and the other is fed the inverted input, so the output of one amp is 180 degress out of phase. The speaker output is then taken from the two output terminals. The common is not connected.
The voltage gain may have to change, but more importantly is that SOMETHING limits the output power. (V p-p available, R (the speaker Z) or I)
With P = (I^2)*R, I can limit the power. So, while any "bridge" configuration could double the voltage, I^2 available must be high enough to change the power assuming the same load.