The school form type Blank Form will update user profiles when the Custom Fields - Publishable block is filled in by the user.
The custom field will update in the profile of the user. The user can see this information as long as the Profile Fields section is setup for their role to view these fields. To make sure this is set up correctly, navigate to:
Core > Settings > Profile
Confirm the All School dropdown is set to the appropriate role
Confirm custom fields are selected to be viewable or editable