I'm trying to update a paid app, bought by a family member but shared with family sharing, on the Mac App Store, and it prompts me for the password. The Apple ID is the family member's, and greyed out so I can't change it. I don't think we're supposed to share all our passwords in order to use Family Sharing, right?
TL;DR - It's possible you have a problem with date / certificates or Apple messed up server side - contacting support to figure out a workaround is likely the "answer" and you probably didn't make this happen.
This is an aggravating bug and it has been around for quite some time (from iOS 9 to iOS 10 is when I first experienced it).
I'm certain it's a bug since I never, ever have used the password for the other two members of the family that buys apps that I then installed using the App Store and my ID / password. Also - if this was broken for everyone, all the time, it would be explained in a KB article or fixed.
I have yet to have anyone from Apple support end up with anything other then ¯\_(ツ)_/¯ and I have to delete the apps and reinstall them.
I hope someone else can get more details or point out something I've missed and even better, I hope I can come back and edit this to say it's fixed when that day comes.