(This isn't a perfect analysis, because the only codified data that's available is how many students declared each department as a double major, but it doesn't say which department they started off in.)
Last year, 24 seniors had declared a second major in physics, and 29 had declared a second major in math (second major data
here). 85 students graduated with physics degrees, while 102 students graduated with math degrees (degree information
here). So at least ~30% (and likely more) of the students in each department graduated with a degree in another department.
Your son sounds like someone who would have a great time at MIT, although I can't speak to the Princeton side of the debate.