customer commerce profile = user profile
You've got some ways to do this, I'd look first to Commerce Extra module which allows you to have an address in the user entity and prepopulate information in the commerce checkout with this.

If you want to actually duplicate the information and have a customer profile entity for each user registered, you can use Rules module for this (it is required by Commerce), you can then create a new rule, for instance in the "After saving a new user account" event that triggers an action "Create a new entity" of type commerce customer profile and then another action, "Save entity".
You can check Drupal commerce default rule "Create a new account for an anonymous order" for a similar concept.