There are different email addresses you can add to an Apple ID. Here are the different types in a nutshell.
Primary Email Address
This is the email address for your Apple ID to login and normal communication. Everybody having an Apple ID has a primary address.
Alternate Email Address
Alternate addresses are optional and can help people find you on Apple services. Most people won’t need an alternative address.
Rescue Email Address
Rescue email addresses are used for accessing your Apple ID when the primary one isn’t working for some reason. This is a security measure and might help you. It is optional and only available if two-factor authentication is disabled and security questions are being used.
Notification Email Address
This address is only used for notification purposes if you have two-factor authentication enabled and replaces the rescue email address.
If you get the error
Your new Apple ID cannot be the same as your notification email address it means that these have to be different email addresses. The point of the notification email address is to be different than the normal one for security reasons. All those addresses can be changed when managing your Apple ID.
Apple has a thorough explanation in their knowledge base for these different types of email addresses.