Even though not all universities in Canada admit straight to the major, students aren’t just admitted to a university in general either. At a minimum they are admitted to a faculty. Similar to the system in the U.K., students applying to university in Canada are required to present with specific grade 12 university level subjects (with the exception of those in Quebec since they don’t have grade 12 instead they need CEGEP courses). What those subjects are will depend on the programs they are applying to be admitted to. So no, not all students are required to take pre-Calculus and Calculus, just as they aren’t all required to take science subjects either. The only course that it is mandatory for all students to present for admission to university in Ontario at least, is university level grade 12 English (or sometimes French). What other subjects students take and the level they take them at depends on the university program they intend to pursue. For those students who are undecided they generally take a wide range of courses to keep their options open and that will often include math up to Calculus and at least 2 sciences. Those who are certain what programs they want to pursue can be more targeted in their grade 12 course selection.
Students applying to programs in the Arts and Humanities are usually only required to present with grade 12 English (or French as the case may be) and the remaining courses can be chosen from among any acceptable grade 12 university level courses. Students applying to direct entry professional programs (engineering, nursing, business etc) and those applying to programs in the faculties of science and social sciences will have more requirements in addition to English. While not all of these programs require students to have completed Calculus, the vast majority do. At a minimum they’ll probably require pre-calculus.