Typically the first year courses for math and engrg majors start with calculus. The Engrg. majors take more courses in the freshman year having to do with engrg. So if you start with math major first, it’s harder to switch to engrg b/c you have to take all the freshman engrg. courses. Math is usually in the college of liberal arts, so if you switch out of engrg, you just have to maybe catch up on some liberal arts electives, which are much more flexible.
I think astronauts come from all sorts of scientific backgrounds. I don’t think it’s the sort of thing you apply to right after graduating with a BS degree. I am assuming she has checked out the NASA web page for astronaut requirements? They take only a handful of trainees out of thousands of applicants.