When you look deeply you will find that Contacts are neither stored on an iPhone nor on SIM. They are synced to an email account that you may have added and enabled Contacts. It could be Google, iCloud, AOL, Exchange, Outlook, Yahoo or any Other email client server.
When you enable Contacts in any of the email accounts they are displayed to you through the Contacts App. When you edit these contacts on an iPhone they are ultimately edited on the client-server.
Check if the email account that stores contacts have also been added to your child's device. Either remove the email account or disable Contacts from the Email Account.