Manage social user details in Google Flutter application

Your proposed solution is good enough. The returned id is unique for your app.

Use the id returned to your app as the identifier for users in your app.

Checkout this answer on unique id

Just leaving the social_login plugin here

// Import package
import 'package:social_login/social_login.dart';

// Instantiate it
 final socialLogin = SocialLogin();

//Before calling any methods, set the configuration
socialLogin.setConfig(SocialConfig(
      facebookAppId: FACEBOOK_APP_ID,
      googleWebClientId: GOOGLE_WEB_CLIENT_ID, /*In case a Google tokenId is needed*/
      twitterConsumer: TWITTER_CONSUMER_KEY,
      twitterSecret: TWITTER_CONSUMER_SECRET,
    ));

// Get current logged user
 final FacebookUser facebookUser = await socialLogin.getCurrentFacebookUser();
 final GoogleUser googleUser = await socialLogin.getCurrentGoogleUser();
 final TwitterUser twitterUser = await socialLogin.getCurrentTwitterUser();

//Log in social networks
 final FacebookUser facebookUser = await socialLogin.logInFacebookWithPermissions(FacebookPermissions.DEFAULT);
 final GoogleUser googleUser = await socialLogin.logInGoogle();
 final TwitterUser twitterUser = await socialLogin.logInTwitter();

//Log out from social networks
 await socialLogin.logOutFacebook();
 await socialLogin.logOutGoogle();
 await socialLogin.logOutTwitter();

By using the built-in sign in using Firebase Authentication SDK (Official Guide), you will be able to obtain login information for different platforms like Google, Twitter, Facebook and even Github.

You do not need to code your own login/signup, everything is handle hassle free by using the right function from Firebase.

After that you will have the login information like so (this is from email login but Facebook, Google, Twitter and Github is available):

You can then link either the identifier or the User UID to the information in your database.

Oh and last but not least, do remember to set the persistence setting for the login provider to keep user login after they close their apps.

You should split Flutter app into 2 parts:

1. The "main" part of the app. It shows user related data.
2. Sign in/Sign up part.

The logic is quite simple. On the app start you check if the user is authenticated. If yes, navigate him to the main screen. If not, present him a sign up screen. Here's how it can be achieved using Firebase as a backend:

final currentUser = await FirebaseAuth.instance.currentUser();
if (currentUser != null){
  // We're good: the user is authenticated.
  // Show him the main screen.
  Navigator.of(context).pushReplacement(MaterialPageRoute(
    builder: (context) => HomeScreen()
  ));
} else {
  // The user is not logged in.
  // Show him the sign in/sign up screen.
  Navigator.of(context).pushReplacement(MaterialPageRoute(
    builder: (context) => SignInScreen()
  ));
}