I think you will see the same thing for many other types of databases, but they will all be dependent on type. The databases already there are music, contacts, notes, PDFs in iBook, epubs in iBooks. I think you'll see the full ability to access and modify those databases, and full cloud syncing of them as well. But, they will still be different and separated by file type or task.
Apps with their own document type register as such with the OS. The "Documents" app shows a list of those apps and each one leads to a list of documents for that app. From this list, users can open, rename, delete, attach to an email, and so on. Apps would not be allowed to dump internal junk in their document folder, only stuff the user creates and knows exactly what it is.
As long as this model was enforced by the OS and by app reviewers, it should be easy for anyone to understand. It would give iOS something fundamental that Android doesn't have at all, which it really needs because it's playing catch-up with most other features right now.
1. The user can manage all their stuff in one place with consistent functionality across all apps.
2. The OS can implement generic file handling features like search, share, backup, sync, etc.
3. It paves the way for standard file types that can be handled by multiple apps, or apps that can work with any type of file, like Dropbox.
The power of filesystems is well understood and I'm sure Apple could create one instilled with their magical UX and just-worksiness. But they won't do it because I think they are afraid of 3rd party apps becoming too useful. They don't want anyone else creating an ecosystem that they don't control. But Google and RIM are happy to do that and I hope they establish a higher standard of utility that Apple is forced to live up to.