Alternative for Storage permission
It's a little more subtle than that. The Storage permission does NOT give the app the ability to read and write everything on the device. It gives the app the ability to read and write things like SDCards and USB devices. It does NOT give the app the ability to read and write everything on the device's main storage (where apps put their databases by default).

The reason it doesn't restrict to the app's own files is that external storage is generally formatted with the FAT filesystem, which doesn't have any concept of file or directory level permissions. There's simply no way for Android to enforce the restriction of files. Further, since such storage is often removeable, even if the filesystem supported permissions, there would be nothing stopping you from taking the media out and using another computer to change said permissions.