MC database lives, by default, under the user profile folder, in %AppData%\J River\Media Center 30\Library. It's a per-user app. When you start MC in the Family user it does not see your DB; you need to create one for that user, or connect to an MC Server library - this is what it means when it asks you for the AccessKey/URL. You can either create a new DB (point it to an empty folder) or connect to an existing *networked* library.
Sharing a local DB for multiple users on the same machine is not supported, I think. There are several workarounds, none ideal:
1. on the main user, move the DB to a common folder outside of the user profile and grant NTFS permissions for both users to the folder. Then just add that Library to the MC on both user accounts. The problem here is that both users will be admins, and you cannot have MC open on both accounts simultaneously.
2. Enable Media Server on the first account, then login to the Family account and connect to the library on localhost:52199. This allows you to set the Family account as readonly, but requires that you first login into your main account to start the Media Server before the Family account can work; both accounts need to be logged in (caveat: not sure if MC supports this)
3. Add a login script to the family account that quickly overwrites the Family MC database with your Main account MC database. Then it just works, they can do whatever they like to the DB as it will be overwritten on next login. The minus here is the added delay on login for the DB copy, but that should be only a few seconds on an SSD. You can also do this by replacing the MC startup shortcut.
The ideal solution is, of course, to have a second PC on your network holding the DB and acting as the MC Media Server, and then both accounts on your main PC can connect to the shared library (one as admin, one as readonly).