What steps are needed to remove or change the notification email?

From: About your Apple Account email addresses - Apple Support

Add or change a rescue [for most people called a notification address] email address

1. Sign in to https://account.apple.com .

2. Select Sign-In and Security.

3. Select Rescue Email. You might be asked to answer one or more security questions or enter your Apple ID password before you continue.

4. To add a rescue email address, select Add email. Then enter your rescue email address, then select Save changes. We send a verification to that address.

5. To change a rescue email address, enter the new address. We send a verification to that address.

6. Enter the verification code and click Continue.

If at any stage you don't get an expected verification see: If you didn’t receive your verification or reset email - Apple Support

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 websites and apps on iPhone - Apple Support

That message generally means that you can’t make the change via the web interface; it MUST be made using of one of your Trusted Devices.

That being said … I’m not aware of ANY way to set the notification e-mail OTHER than via the Account web interface.

Were you using a Safari browser in/on one of your own Trusted Devices … or a browser on “somebody else's” hardware ???

Thanks for the feedback.

I always use Safari from a Mac when working in account.apple.com so I’d not encountered THAT particular “alignment of the planets”