Unable to change notification email address for Apple ID

The "Continue on one of your devices" message is likely part of the implementation of the Passkey feature for iOS, iPad 16 and Ventura. It is currently only supported by certain web sites and apps, including the Safari app. The site or app is making a call to your trusted device(s) for you to authenticate from the device itself (using Face ID or Touch ID). Tap Settings, then your name at the top and it should prompt you for Face ID or Touch ID, then approve the action on the other device. Using passkeys is described in this support article —> Use passkeys to sign in to apps and websites on iPhone - Apple Support

If you are still having problems with this feature when using Safari then an alternative solution is to not use Safari for what it is you are trying to do. Use a different browser that does not have this feature in it. See this August 17, 2023 post by Community Specialist Chris_V89 at i followed the exact steps of the app spe… - Apple Community

"does this issue occur […] at the Apple ID website in Safari? If so, to help with troubleshooting the issue further, try testing the issue in another web browser, if possible."

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."

- "Don't use Safari or Chrome. I got it to work using Firefox on my Mac."

- "the Chrome browser works for me."

I'm trying to add a new app-specific pass… - Apple Community

- So I ended up getting on the phone with Apple about this… turns out it was an issue with Safari. As soon as I tried to create a new app-specific password in Chrome, it worked perfectly.

How do you create an app specific passwor… - Apple Community

Using Chrome did the trick for me.

Can't create app specific password - Apple Community

Wow-using chrome worked

Can't generate App-specific Password on i… - Apple Community

I used my Mac, used Brave as my browser, and the process worked like a charm

Can't Change Recovery Email Address - Apple Community

- “Very similar resolution as well.”

- “As soon as I switched to Firefox it all worked perfectly”

This is a user to user forum. Apple does not respond here (not since last October). Contact Support.

Refer to this page for Apple Support features ➔ Contact - Official Apple Support

Select from the presented options until you find a solution for your issue, or see if there is a chat or phone call contact method offered lower on the page (you sometimes have to narrow down the options multiple times before this is shown). If you do not see your issue, keep experimenting with any series of selections until you reach one that offers a chat session or a telephone call and get the representative to redirect you.

This page has telephone numbers for contacting Apple for support and service ➔ Contact Apple for support and service - Apple Support