If you click through to the Privacy Policy it states:
Individual choice: Submission of your personal information to this website is voluntary. If you choose not to submit your personal information to this website you will be required to complete paper forms to submit a college admission application, financial aid application and other student services applications.
So in a nutshell, it sounds like if you do not give consent, you’ll have to fill out a paper application, probably in person.
We enrolled our S25 last summer (in a CCC) and I don’t recall ever completing this. If I remember correctly there was a special application for high school students. Why would you give your own SSN if it is your child’s enrollment? I think I did have to give consent as he was under 16 at the time.