If you want to create an Apple ID for a child under 13 years old you have to do it on your own Apple device in Family Sharing. One disadvantage in doing it this way is that you are forced to create a new @icloud.com email address and use it as an Apple ID / iCloud ID for your child's account. So if your child already has an existing email address he / she won't be able to use it as an Apple. To make things worse, if you created an Apple ID with @icloud.com email address it is no longer possible to change it (you can do it with no issues if you used a third party email address).
Luckily there is a workaround that allows to create a child account for under 13 year old while using any non Apple email address as an Apple ID:
- Go to www.icloud.com and click on Create yours now.
- Create a new Apple ID using any email address you want, but choose the birth date so that it puts the child between 13 and 18 years old (you will change this later).
- On your iOS device (iPhone / iPad) go to Settings > [your name] > Family Sharing > Add Family Member and invite the child to join your Family Group using just created Apple ID.
- Login to child's iOS device with his new Apple ID and go Settings > [child's name] > Family Sharing. Accept the invitation and join your Family Group.
- Now go to appleid.apple.com, login with your child's account, in Account section click Edit and change the child's birth date to the correct one. You'll receive a warring that this action will convert the Apple ID into a child account and that Family Group organiser will have to approve this change (Apple will send you an email with a link to confirm the change).
- Login to your own email and follow the link to accept date of birth change.
- That should be it. You now have a child under 13 year old in your Family Group and it has a non-Apple email address as the Apple ID.