The only success I've had (and I haven't tried it for all apps, so no guarentees), is to delete the app and then, instead of downloading it from the search tab or the store tab, going to the following location:
Updates --> Purchased
It's at the very top of the screen and once you click on it, you are presented with a list of family members. If you select the family member who actually purchased the app and find the app in that list (search box at the top too), then you can re-install the app from there.
It appears that the issue, at least in the case with my wife and I, is that the app was originally downloaded to her phone using my apple idea (when we shared the ID). Now that she has her own, we have to effectively re-download the apps in the above manner to tie them to her Apple ID. I royal pain that I hope they fix soon.
I don't know if this works with apps that are not setup for family sharing by the app designer as referenced earlier in this thread...
Edited to add: Note: This DOES NOT work with apps that don't support family sharing, at least for me. Following the above instructions, the app will not appear in the list of apps that family members purchased, but when you search for the app in the main app store, it doesn't present you an option to download it using the current Apple ID, but instead gives the cloud icon showing that you've already purchased it. Trying to download it again gives the same pop-up as the original poster. This is seriously broken.
Edited to add 2: It does appear that, after deleting the app, logging out of the iTunes store and logging back in does appear to fix the problem of apps that don't support family share. Still not idea, but workable at this point.