How can I remove an obsolete email that is being used to notify of security risks, it states to go to another device?

Read this article about managing the information you use with your account --> Manage and use your Apple ID - Apple Support

go to the section about Add additional email addresses to help people find and communicate with you on Apple services like FaceTime, iMessage, and Find My.

This "continue on" message seems to be appearing in various contexts when editing Apple Account information with a web browser. You can try editing the information as indicated in the messages by using Settings. Others have found using a different browser or simply trying again may work.

Related discussions:

every time I tried to create an app-speci… - Apple Community

"Someone in another post said to use a different browser (not Safari or whichever one you normally use). That worked for me."

"Use a computer to generate the App specific passwords. Don't use Safari or Chrome. I got it to work using Firefox on my Mac."

"the Chrome browser works for me."

Can't change Apple ID - Apple Community

"OK it just worked. No idea what was going on before. Thanks for the help anyway!"

Can't Change / Remove Apple ID Notificati… - Apple Community

Occurred while changing account information using a web browser.

Click on this link for Apple products feedback links --> Product Feedback - Apple

Apple may not respond but your comment will be read.