Both Chrome and Firefox disable screenshots in private mode. I assume that this helps defend against 3rd-party programs running in the background that might save screenshots unbeknowst of the user.
In older Androids it was too easy for the app to get the permission to capture/record the screen without clear user consent. In newer Androids you need to ask permission at runtime.
It's only a part of the issue. The other part is that a "launcher" app shows miniatures of apps when you scroll through open apps, and this behavior is controlled by the same screen capturing permission. When you enable screenshots, you also enable previews in the launcher app when switching between apps, in some cases it might be not desirable.
There's probably some more nuances that I'm not aware of.