I have created a new forum hosted on Discourse itself. The forum is for members of an organization. I want to use the Full name requirement (and I have that selected under Settings > Users > Full name requirement.
But, when a user tries to register he/she gets redirected to a Discourse ID login which allows them to sign up only with a username.
I think you are running into a different issue. I went through the same flow, and I can reproduce what @hammerdunner showed in the OP, the new user gets logged in straight. I also notice here that ID autofills the full name field.
We need to fix this, I’ll move this to the Contribute > Bug category.