Hi,
I already understand all that.
And as I said in my two previous messages, you're in the case where
I believe it has something to do with either the file templates/YOUR_TEMPLATE/html/com_hikashop/user/registration.php (an override of the view displaying that area) which you might have on one website and not on the other, or they have different versions, or you have the "use boostrap design" setting configured differently on both website (resulting in using "registration_boostrap" instead of "registration"
Do you have that view file ? If you rename it in order to deactivate it, does it help ? How is the "user bootstrap design" setting configured ? And if you change it does it help ?
So please check on that. If you don't, we won't be able to move forward here.