Some things to consider:
- Your database architecture has absolutely zero impact on your clients and vice versa. An API sits between these two layers and should handle all translation between them.
- The method of authentication is entirely up to you. Firebase and such support out-of-the-box oauth with Facebook, Google, etc. But you don't have to use that, you can store a username/password combo and be done with it if you want.
- Realise that what you're trying here is not just a big project, it's also a very complex one with many layers. You're basically doing full-stack work here, which sounds great on a resume, but is really not something you'd normally do by yourself.
Now for the solution. What you want is to make a choice, do you want to host your database yourself or do you want to use a platform like Firebase or Azure? If you want to host yourself you'll have to look up VPS prices and find the best option. Then get a VPS, install MySQL and a database manager, create some tables and setup a backend. If you want to use a platform like Firebase, compare them and take your pick, it doesn't matter a whole lot. Then you do the same as with a VPS, but with simple tools in an admin console.
Your first endpoint should be a register. You provide it with a username and password and whatever else you want to store about your user and it saves it in the database. Then you make an authenticate endpoint that verifies a user's credentials and gives back a token to use in subsequent requests.
Once all that is done you can start hooking up your apps, let them make https requests to your API and see the magic happen.
Personally, in your situation, I'd just go with Firebase right now. You can host a database there, setup a simple API to let your clients access it and even implement analytics and crashlogs etc very easily. You'll skip all the headaches of scaling up a server's storage capacity, optimizing the database, creating logging and monitoring pages and managing third party tools like your database, database manager and backend language. You don't have to use all Firebase's tools, but it's nice that they're there and just an SDK include away.