There is an option to add School name and other education fields on the New User Registration section of the User Login part and a Profile Form.
You notice the School name field appears as a required field and cannot be deselected. You may also notice when editing the part, you unmark School Name and click save, but when reopening the part, School name is marked as Include again.
It seems as though you not able to deselect the School name field.
Parts that includeeducational fields for example, Class of, force the NetCommunity form to also include and require School Name be included. The Raiser's Edge has a requirement for educational records to have a school name if any education fields are included. NetCommunity has made this a requirement to match the Raiser's Edge.
If any education fields, under Primary Alumni or Education Attributes, are included in the part, School Name must also be included and Required.