So I am applying for the Walgreens pharmacy internship program and on my application, it automatically says "Pharmacy Intern 3". Does that indicate I have to be a P1 student (in the 6 year program) or a P3?
That indicates that you're a P1 (in a 6 year program). It doesn't really matter what year it states other than possibly your pay rate (which is determined by your district supervisor anyway).
I go to a 4 year school and as a P1 I was hired as a "Pharmacy Intern year 3" in Winter. In June, I automatically got pushed up to "Pharmacy Intern Year 4" and my hourly rate got bumped up.