Ok internet poking did not yield any results (could also just be using a different description of this issue than anyone else might). Nonetheless, I may have corrected the issue so wanted to leave this here in case anyone came across this in the future:
I had thought the apps in questions were downloaded by going to the App Store > clicking name in lower left > changing the purchase toggle to the correct family member > clicking the download icon (a way I believe works without issue). Instead, my new impression is that these apps were downloaded when that user had an account on this Mac.
The Apps appeared for all users on the Mac to have access to, but they were linked to that specific users ID. As a result, even once the user account was removed from the device, the app was still requiring that user's ID to sign in again because that is what the license for that computer had been purchased/downloaded under.
To fix this issue I deleted the app (making sure all save data was somewhere it could be found) and then re-downloaded it using the correct Family Sharing app downloaded process I wrote above (How to download apps and content that your family members purchased - Apple Support). I will need to verify this on a future update but believe this has solved the specific issue.
