The document profile and monitor profile are different things.
The document profile such as Adobe RGB 1998 or sRGB IEC61966-2.1 describe what actual colours the colour values in the document represent.
The monitor profile describes what colours are displayed by your monitor when sent colour values. It is specific to your monitor in its current state of adjustment. So changing the brightness or contrast requires a new monitor profile to be built.
Using both profiles, The colour management system can translate the values sent to your screen in order to display colours correctly.
Now screenshots. Taking screenshots is not the best way to move images in and out of Photoshop, but if you must use a screenshot do this:
1. Take your screenshot.
2. Open it in Photoshop
3. Assign the monitor profile (remember there is only one valid monitor profile).
4. Convert to a document space such as sRGB IEC61966-2.1
5. You can now use or export the screenshot.
Dave