Stanford has its own post office, or at least it did when I applied there. I was actually born at Stanford, but that fact didn't cut any ice with their admissions board.
Looks like you're right. I used to live in Palo Alto and never heard of Stanford, California. It's not listed on Mapquest as a California city. My son was born at Stanford University Hospital, which at that time was in Palo Alto, California. But on the Stanford website, I find this:
What is Stanford's address?
The primary address for Stanford is
Stanford University
Stanford, CA 94305
Maybe Stanford, CA was invented after I left.