Can someone request another security key or change it without your knowledge

Note that there is a difference between a security key and a recovery key.

About Security Keys for Apple ID - Apple Support

This is what somebody would have to do to set up a security key:

On iPhone or iPad

1. Open the Settings app.
2. Tap your name, then tap Sign-in & Security.
3. Tap Two-Factor Authentication.
4. Tap Security Keys, then tap Add Security Keys. Follow the onscreen instructions to add your keys.
5. Review the devices associated with your Apple ID, then choose to:

• • Stay signed in to all active devices.
  • Select devices that you don't want to continue to have access to your account and sign out of them.

To stop using security keys: Open the Settings app, tap your name,

then tap Sign-in & Security. Tap Two-Factor Authentication, tap

Security Keys, then tap Remove All Security Keys. If you remove all

security keys, your Apple ID reverts to using the six-digit verification

code for two-factor authentication.

Read this support article for requirements if you are using a recovery key:

Set up a recovery key for your Apple ID - Apple Support

unless that other person has Access to your unlocked phone, This is not possible