This is true for any bridged amp. Bridging + 4 ohm speaker = trouble.
No, not every bridged amplifier. For example, the Carver M-400 is 2 ohm stable in stereo mode, and can be bridged for use with 4 ohm speakers.
In general, the minimum impedance an amplifier can handle bridged is simply double the minimum impedance it can handle unbridged.
When I think about that Carver M-400 amplifier, which, I believe, was the first product of the Carver brand, it is no wonder that it caused such a sensation when it was released. A relatively inexpensive 201 watt per channel (into 8 ohms) stereo power amplifier, that is a small cube, and is stable into 2 ohms, and can also be bridged, that is something quite remarkable.
Many of them are still going strong after about 30 years of use.