I'd like to request to use $form_values['mail'] instead of $form_values['name'] in function bio_user_register_submit($form_id, $form_values).

This will solve a problem with the "Email Registration" module, which changes the username during the registration process, so it can't be used as a key when the uid is fetched from the users table.
The result of this problem was, that the uid field for an entry in the bio table was always 0.

I've changed it on my site and from what I've seen it works. I guess the email address of the account needs to be unique as well and it's only used for a one-time-action at registration.