Current password required to change password for self-registered users. No password was ever set
I have solved my issue by re-installing the login one time module and clicking the Ignore the current password field checkbox.

This hides the Current password field for the one time login link, and everytime after requires the user to provide a Current password if they want to change their email, password ect.

The problem is that this website will still ask for a current password for validation links if I uninstall the Login-one-time module, which makes no sense to me.