it’s not illegal to put account deletion behind a login at all. Its also legal for them to request identification.
However if you request data deletion and they have no valid exception to avoid doing that they must comply, it doesn’t matter if they have a mechanism for deletion that you can use, they have to still delete the data even if you don’t press the “delete account” button.
you can file a complaint with your countries regulatory departments but if they refuse to press the delete account button for you, there’s not much else you can do outside of that.