(Source: I am a math professor at Stanford, with connections to the CS department and many former students in the CS Master's program, and experience with graduate admissions.)
Nothing about this situation suggests that you are any less intelligent than your classmates. The admissions committee isn't running a charity: if they thought you weren't prepared to excel, they would have admitted another student from MIT instead of you. They invest a lot of effort into determining which students are positioned to be successful; it's not a perfect system by any means, but I think they know better than you do whether you're prepared.
I have never considered undergraduate institution or background when considering who to work with, and I haven't heard of any colleague doing so either. As a grad student you get better and better at learning, so it's easy to patch any holes in your background.
If you are concerned that you haven't had the opportunity to take all the classes that the CS undergrads at Stanford have: well good news, the majority of CS courses are online, including all the lectures, slides, and homeworks. And you have 7 months before you'll start grad school this fall -- that is plenty of time to complete all the courses you're worried you've missed, if you really feel that's the best use of your time.