When I’ve tried stereo/mono bridge-able Solid State, the bridging brings a lot more power - up to 4x power, if the amp truly "doubles down" because its PSU / output stage / heat sinks can handle the strain. But more often you’ll actually see a ~ 3x power gain or thereabouts. HOWEVER this approach is not without sonic downsides. It’s turned me off of bridging. Not only are you running the amp much harder, but the bit of bridging circuitry itself may be a less than ideal implementation. And yes each channel now "sees" only half of your speaker’s impedance - so if your speakers are nominal 4 ohms, your amp would need to safely drive 2 ohms (lower for dips, so it should be stable to 1 ohm).
The switchable stereo/mono tube amps I’ve had (VAC) use a different approach than bridging. They parallelize the 2 stereo channels at the output transformers. This approach "only" doubles output power, BUT the sonic impact is ALL positive - no downsides like with bridging. This approach kind of has the opposite impedance problem of bridging. The output "taps" will shift around. What was formerly an 8 ohm optimized tap (stereo mode) is now optimized for 4 ohms, due to the parallelization. So you'd need 16 ohms stereo taps to run "optimal" mono into 8 ohms. VAC doesn't supply those 16 ohms taps, and my speakers are 8 ohms. But it still works great :)