So we all know V = I * R.
Bridging a 2ch stereo amp to mono doubles the output Voltage, halves the load Resistance each amp channel "sees", and then because of this each amp ch also *tries* to supply double the current (I), but how much of that it can actually supply depends on the design of its PSU and output stage.
The voltage is doubled by having one channel amplify the +V, and the other ch amplifies its inverse -V. Thus the difference makes V - (-V) = 2 * V, though each channel by itself does not see more than 1 * V in magnitude. However, each channel does try to double the output current, because they are each driving half of the same load (one 8 ohm speaker), rather than each driving two wholly different loads (Left and Right 8 ohm speakers).
The inversion for the -V side necessitates some kind of circuitry to do the inversion. This itself is not always without distortion artifacts. And the fact that the amp (both channels) are now operating into half the impedance load will also affect its distortion products, and thus sound. So if you value stereo image symmetry, it’s not great to run one side with a bridged amp and the other with half a stereo amp.
As for the power output of a bridged amp -- if you have an amp which "doubles down" its power into halved impedances, then you will net 2 * V and 2 * I, and Stereo Power = V * I, so you get Bridged Power = 2 * V * 2 * I = 4 * Stereo Power. Fully 4 times the power! Of course, it’s no free lunch. You’re driving the amp much harder and hotter in the conditions. Almost all amps, even "high current designs" will show their current limitations to some degree, and thus will not rate a full for a 4 times power factor.
I’ve tried bridging with a pair of Solid State stereo amps I really like (Phison A2.120SE). I found that their sweetness which I enjoyed in stereo mode largely disappears in bridge mode. So I kept them in stereo mode. It was better to run them in either bi-amping or single amp stereo mode configurations.
I also have a pair of stereo tube amps with mono mode, but in this case the 2 channels are paralleled, not bridged (it goes this way because of the output transformers). It only doubles the power (at maximum), but unlike bridging it only improves the sound quality!