I'm only familiar with the University of Wisconsin system. They use the math test to slot you into levels ranging from calculus to a remedial math course. Don't know why they don't just use SAT/ACT.
As for majors, there are limited admissions into some programs such as engineering and business, where you are not formally admitted until after your first year, and have to make a certain GPA. Not all majors had such requirements.
Then there were some unique situations. Majoring in music performance required an audition, and the incoming students tended to be playing at a very high level.