Remember they also look at the school profile , which will have how many student receive free or reduced lunch. Admissions knows which student have fee waivers for application fees. Princeton is also a Questbridge partner school, they will know students applying through Questbridge. There are plenty of ways to gauge low income without actually looking at financial aid forms. Parent educational level and employment are on the first page of the common app (this can also signal first gen or economically disavantaged students).
I am also sure that the regional admissions officer as a pretty good grasp on the schools in their territories.
You also check the box on your application stating that you are applying for financial aid. At that time admissions does not know if you need a little aid or the whole ball of wax.