Note: The available options vary, depending on the current settings for allowed communication during screen time.
If you want to let your family member edit their contacts, turn on “Allow contact editing.”
If you want to manage a child’s iCloud contacts, turn on Manage [child’s] iCloud Contacts.
To manage a family member’s iCloud contacts, click Edit, then add, remove, or edit contacts.
You don’t see the Edit button until your family member accepts the request to manage their contacts. Until then, Pending is displayed. See Turn off Screen Time on your Mac