PhD application interviews on same day for different programs
It's common. Departments aren't naive enough to think that they are the only place you've applied to. If you email them in good time and explain that you have another interview, I'm sure they will happily arrange alternatives. This could be a visit at a different, mutually convenient time, or perhaps a Skype interview.  Just phrase your email a bit more professionally than "yeah sorry I'm interviewing elsewhere" (but I'm sure you know this).

I would sincerely hope that no university is petty enough to put a black mark next to your name simply due to a clash of schedules -- but if they did, you don't want to work in such an environment anyway.