Assigning a Child Apple ID to a device

You can't add an iCloud account to a different Apple ID than the one it is tied to or is itself.

iCloud accounts are Apple IDs unto themselves so cannot be part of a different Apple ID as a secondary email address or anything like that

As explained before, to add it to a device, you would need to sign your account out of the device and sign in with the new one you just created.

If you need to then supervise or restrict the account, you can do so with Family Sharing.

Link-> Set up parental controls with Family Sharing on iPhone - Apple Support

How are you trying to assign it exactly?

Have you signed in with it on the device in Settings->Account(top of settings), Settings->Media & Purchases?

To change the birthdate if its set to something over 13 years old, follow the steps here: link-> Update the birth date associated with your Apple ID - Apple Support

To change it if its less than 13 years old, you will need to contact Apple Support: click here-> Contact - Official Apple Support