Medical admissions is a small field. It would not surprise me a bit if admissions directors were at least familiar with each other. There are professional conferences and organizations dedicated solely to medical admissions. It's pretty insane, but they exist. As another example, I know at my school admissions directors from a small number of schools are invited to the campus for a day to do a mock interview day of sorts. As you might expect, the pre-health directors from all the Ivies were there as well as some other individuals.
As far as it relates to this case, letters of interest are appropriate but sending multiple letters of intent is poor form. As you progress through your medical training, the field of people that you can get "lost" in gets steadily smaller and smaller. You should avoid doing potentially shady things like that because word will get around and you never know who might hear it. I don't think that's the sort of thing you should be worried about in this case, but again, it's just poor form.