With Windows XP, you could activate fresh installations of Windows using that product key 5 times before activation would fail.
If the key was displayed in System Information, a person could walk by your computer, press Windows + Break to bring up the System Information panel, and copy the product key, with no way for you to prevent its future use.
Viewing product keys in any piece of software can usually be looked down upon as a security issue. I'm not aware of any games or pieces of software that will reveal the key used for activation after the fact, and normal users generally do not have multiple product keys in the way you do.