Solution 1:

When the user is signed in, they get an ID token that is valid for about an hour. If you set a custom claim, their (server-side) profile is updated immediately, but their ID token is not auto-updated. So you'll need to refresh their ID token to get the new custom claims.

As far as I know this ID token is only refreshed by calling getIdTokenResult if it has expired. If that's the cause, calling user.reload() and then getting the ID token should give you the updated claims.

Solution 2:

For me it simply worked taking the advice from one of the comments:

// --------
// Frontend
// --------

// Triggering the cloud function
const url: string = 'url-to-your-cloud-function'
await this.http.post<unknown>(url, {}).toPromise();


// After cloud function was run and custom claim was set -> refresh the id token
// The 'currentUser' is a reference to the firebase user
await this.authService.currentUser.getIdToken(true);

// --------
// Cloud Function - createSubscription
// --------

const createSubscription = () => {  
  await admin.auth().setCustomUserClaims(userId, {
    subscriber: true
  })
}