Understanding the problem on Android
Ironically, due to the intense sandboxing on iOS, there's a straightforward way to make this happen there (App Groups) provided the apps that need to share data are all by the same developer. Likely because Android is more flexible on security, this actually ends up being a more difficult problem there. The Android team have so far not seen fit to provide a convenient and secure way to specifically share this kind of data because there's a low security workaround.
That said, there are plenty of ways to share data between applications without involving the cloud.
SharedPreferences
The original question states that SharedPreferences are deprecated. This isn't true, as far as I can tell, however the MODE_WORLD_READABLE and MODE_WORLD_WRITABLE contexts are deprecated which makes this approach subject to not working in the future. The mode has been deprecated for quite some time, though - since Android 4.2 (2012). There's no threat in the current Android docs to suggest they're actually phasing it out (sometimes deprecation just means "this isn't a great idea" not "this is going to be removed"). I suspect the lack of a more secure OS-level direct alternative for application data sharing at the settings level is probably the reason for preserving it in a state of deprecation for the last 5 years.
File Access
The simplest and most common way I'm aware of to implement data sharing between applications on Android is to simply request file access on the device and create a shared location on external storage for this data. (Don't be confused by the "external storage" designation - this is just how Android refers to shared data. It doesn't necessarily refer to an SD card.) You give the file a unique name, and you store it somewhere that your apps know where to look for it. Best way to get that path is something like:
Environment.getExternalStoragePublicDirectory(Environment.DIRECTORY_DOCUMENTS)
The obvious problem with this is security. While not deprecated by the OS, it introduces the same problem that the Android docs list as the reason for deprecating MODE_WORLD_* - it's inherently insecure and opens up potential exploits in your application.
- You're placing your information where everything has ready access to
it.
- You're asking for permissions that your app may not otherwise need.
- You're reading files that you can't verify the origin of.
If your application isn't handling any sensitive data, maybe this doesn't matter to you (it might to your users). If you're planning to read data from those files, you should ensure you're providing maximum validation for that data before parsing. Check the size of the file, validate the formatting, etc.
Creating your own service
You could always create a Service or an IntentService. (There are subtle differences between the two, but IntentService is a subclass of Service that runs in a Worker thread while Service interrupts the main thread. IntentService also implements Intent support which provides the most straightforward interapplication communication on Android).
This service has its own private storage, for which it has full read/write access, but nothing else does. This service then provides an interface to receive Intents from other apps, and to return results (as Intents) to those apps. This is an extremely friendly way to implement interapplication data while maximizing data privacy and security of that data. If outlying apps mostly need to request very basic information from a central application, this is your entry-level option.
Implementing a BroadcastReceiver
Along the same lines is the BroadcastReceiver class. Depending on what sort of data you're intending to share between applications, and how familiar those applications may be with your specific approach, this another possibility. Again, you'll be managing the shared data under one application's private storage. Communication is done by Intents, so it's similar to an IntentService - except that applications may communicate with a BroadcastReceiver by issuing systemwide events (that is, they don't need to be explicitly communicating with your app or service - they're shouting out the world for a piece of info, and expecting an answer.)
Creating a ContentProvider
The Original Post seems to misunderstand what a ContentProvider is and how it works. You have to think of this type of item like you would a cloud solution - even though it's local to your device. Every app doesn't need a ContentProvider - they all need to communicate with a ContentProvider, and that ContentProvider maintains, updates and returns data.
This is probably the most "Android-y" solution for this particular usecase and offers the greatest expandability. You implement an independent process that handles data storage and responds to other applications. It is, however, a more evolved solution - and as such may be more a more challenging endeavor. If you need a real database service, rather than a fairly simple request/response type service, ContentProvider seems to be the best choice.