I can only speak to the college my DD attends, but a placement test was used for math, as well as for several other subject areas (off the top of my head, foreign language, music theory, biology...and these were just the few DD was interested in, I suspect it was very common). I suspect it is the norm at most selective places, though if it’s important to you I would check.
The math placement testing was left up to individual students before they had to choose classes. The information was sent at some point over the summer, and most of the placement stuff was done online (I seem to remember DD having to go to a scheduled placement test during orientation on campus, but that was an exception). They also gave placement opportunities based upon AP results (though no credit- this was true for foreign language as well). So kids could take the placement test and to a certain extent, place themselves (I’m sure if there were questions you could ask for advice, but it was pretty clearly outlined). For DD the decisions included whether to take a compressed/accelerated class which essentially combined two classes into one, so even within her level of achievement (which was not advanced beyond her HS offerings) there were different paths open to her. (She did choose the compressed class, and was very satisfied with the pace of instruction- finally!). The breadth of possibilities was amazing, and at least where she is, there were no perceived obstacles, just guidance on what would work best for each individual.
