I've already replied directly to the support request you sent by email, so that I can get further details about your invited viewer who's having a problem.
The default registration form includes a link to " signup for just a username" directly to the right of the prompt to create a blog address:
https://signup.wordpress.com/signup/
For an invited viewer whose email address isn't recognized as a WordPress.com user, the link they receive by email to create an account takes them directly to this username only signup form:
https://signup.wordpress.com/signup/?user=1
Please note that for checking out either of the links above -- in case you'd like to see exactly what an invited viewer would see -- you should first sign out of your own WordPress.com account.