Authenticate custom WP API endpoint with social login (OAuth)
Once you have done #2, the HTTP response from WP will include the auth cookie: see https://github.com/royboy789/wp-api-social-login/blob/master/inc/social-routes.php#L31 in the sample pointed to by the question.
Not sure what HTTP library you are using in the app but if you save the cookies received in the response and just reuse them in all subsequent requests to the server then authentication should be taken care of: you won't need an extra check/access point, as the login/registration automatically logs the user in, and the standard WP cookie checks take care of the verification on new requests.