The new sign-in process for iCloud on the web is already available for iOS 13.
Signing in iCloud on the web is likely to get easier as Apple is reportedly testing Face ID or Touch ID for authentication.
According to 9to5 Mac, the new sign-in process for iCloud on the web is already available for iOS 13, iPad OS 13, and macOS Catalina beta testers.
If you visit beta.icloud.com on a device running the beta version of the OS, you will see a new pop-up asking if you'd like to sign-in using your Apple ID with biometrics. It doesn't require two-factor confirmation.