There is not really way to hide browser database, if you still want offline access.
For my web app, not PouchDB, I use database name prefix with userid.
The trade off between security and easy of use depends on how key is stored. There is no way to have both.
1. The key is in memory, so user can access even if when offline, but it will not survive browser refresh.
2. The key is saved in sessionStorage, so it survives refresh, but not on different instance of browser window.
3. The key is stored in localStorage, so it survives until user logout.
The app may ask user if using public or private computer so choose approriate key storage.