Well, first, there is no reason that both of you can't be admitted to the same program in the same year. Many programs will accept the best X students, no matter their undergrad school, so it's unlikely they will pick between one of you unless the cutoff just happens to lie between you too.
But don't just take my word for it, let's look at some cold hard stats!
The Caltech Physics department lists its incoming class each year and includes info on each new student's undergrad program. (http://www.pma.caltech.edu/GSR/gradclasslist.html
). Here are some examples:
2013: I count 3 people from Caltech, 3 from USTC, and 3 from National Taiwan U.
2012: No repeats this year but a smaller class size too.
2011: 2 from MIPT, 2 from MIT, 2 from Cambridge, 4 from National Taiwan U, 2 from Princeton
And this is only the people that accepted Caltech's offer, not the complete list of those who got accepted by Caltech. I've heard that Caltech expects approximately 30% of its offers to be accepted.
Another example is that my officemate in my current program also graduated from the same undergrad as me in the same year.
So, it might turn out that one of you will get an offer but not the other. Unless you are applying to tiny programs, I would be pretty surprised if the school rejected one of you and took someone else less qualified just because two students happen to have the same undergrad! Instead, if there is a difference in qualification, it will be more likely to be rejected because the qualification cutoff is between the two of you instead of because they only want to take one.
I don't think there is any reason to feel anxiety over this. But of course, in your shoes I would feel the same too, because we are humans! The above reasons would help me feel better, but I also take comfort in knowing that at this point, there is literally nothing you can do to change things. Just give it your best shot and see what happens. In addition, unless this other person is going to be applying to all exactly the same schools as you, this should not be a big deal. Finally, I'd bet the other person is worrying about this as well--how can you say for sure that you'll be the one that "loses" if you are directly compared!