iCloud => Verify your email address.

I'm unable to log in into web-based iCloud anymore. I'm stuck into a loop.

1. starting iCloud
2. Username/Password
3. 2FA Notification on my iPhone
4. Verifying the code in the browser

And than I get: Verify your email address.

=> If I stay with my @me.com address I get the error:

You can only add the @me.com address that is already on file in your account.

Than I have tried a gmail address, and but I have received an Email that I want to change my Apple ID and should confirm!

BUT I DON'T WANT TO CHANGE MY APPLE ID, I WANT TO LOG IN!

What is going on here, has anyone an idea?