Sonny Software

Since this was a change to the way Apple website provides authentication, the option to stay signed in is not provided anymore. We submitted a bug report to Apple. Currently, instead of “Continue with Password” please click on the “Use a different Apple ID” link. The usual Sign In window with the "Keep me signed in” option will appear.

This fix, while not intuitive, works.

I was just about to post my issue with this here, so thanks for pro-actively dealing with this.