Why does the MacOS App Store ask for other users Apple ID password?
Software you buy through the App Store is registered to an Apple ID, not a computer. This gets confusing when multiple people use their Apple ID to install software on the same computer. 

The installers for some applications, often ones that have processes running the the background all the time, are often smart enough to quit and restart as part of the install/update process. 

Hope this helps!
Ric