While the HS will not see your actual FAFSA, they will know if you have filed one as the information is being fed back to the schools to ensure that students have access to college and financial aid.
At the NYC DOE where college readiness is part of your report card, and in NYS, where you need to file the FAFSA in order to get NYS aid, the schools in NYC and NYS do get a report stating what percentage of their students filed a FAFSA. I don’t think they get the actual names just the number of students in the cohort and the number of students who filed-. You will then have to back the numbers out (subtracting out students who are ineligible to file).