Yeah that's absolutely wrong of her to be making those statements.
You are correct on the process. Once a test is flagged positive it will be sent up to a licensed physician (Medical Review Officer) and they will reach out to your and verify whether the positive test can be explained with a prescription. My background check and drug test was done by Castlebranch, they do checks for many schools nationwide. I had to reach out to them after a couple weeks because I was getting nervous that they failed me and never got in touch.
Also if you take meds like adderall for ADD or ADHD our school has to give you extra time on tests and special accommodations. So I am pretty sure she is violating some sort of Americans with Disabilities Act laws here by asking and singling out those students like that.