Can you block an iPhone from adding another Apple ID

Not if you set parental controls to prevent them from editing the Apple ID field.

See how to 'Use Screen Time on an iPhone, iPad, or iPod touch' by clicking here --> Use Screen Time on your iPhone, iPad, or iPod touch - Apple Support

With Content & Privacy Restrictions in Screen Time, you can block or limit specific apps and features --> Use parental controls on your child's iPhone, iPad, and iPod touch - Use parental controls on your child's iPhone, iPad, and iPod touch - Apple Support

It’s good that you’re asking all these questions ahead of time.

A key step starts w/ creating the child’s AppleID.

It must be created by an adult who is the family organizer. See:

Set up Family Sharing - Apple Support

Create an Apple ID for your child - Apple Support

Next the ADULT needs to setup the child’s phone. When setting up ScreenTime, be sure to pick “This is a Child’s Phone” and setup w/ a screetime passcode only you know and is recoverable w/ YOUR AppleID.

Lastly, effective parental controls mean that the child shouldn’t be able to change his/her AppleID, Password, or Passcode (or else YOU loose monitoring access too).

Both of these parental restrictions can be set in Screentime.