Hopefully not anymore. About a decade ago it was revealed that the FAFSA showed all colleges to which you applied and not only that, showed the order you listed them! It was claimed that some colleges were using that order as a proxy for the student preference. This was supposed to be fixed, colleges don’t see who else gets the FAFSA.
An article describing this from Inside Higher Ed is at Colleges use FAFSA information to reject students and potentially lower financial aid packages