This "continue on" message seems to be a new message appearing in various contexts. If you are doing something where an app-specific password could be involved*, follow the steps in Sign in to apps with your Apple ID using app-specific passwords - Apple Support
Others report having encountered this in other contexts such as when editing Apple Account information with a web browser. You can try proceeding 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.
I can't find how to continue App Specific… - Apple Community
- "The solution to the problem is not to allow the iPhone or Mac to sign you in automatically with FaceID or TouchID, but instead I found complete success in manually signing in with my Apple ID and password, which then prompted a two-factor authentication prompt and code. Following that sequence, I was able to successfully revoke and add app-specific passwords."
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* I'm trying to add a new app-specific pass… - Apple Community
