ABET accredited engineering majors must have at least 3/8 of the credits in engineering science + engineering design, 1/4 of the credits in math and natural science, and an unspecified amount of general education (even Brown requires 4 courses (out of 30-32) in humanities and social studies for ABET accredited engineering majors, so it would be reasonable for the minimum to be around 1/8).
So we can assume that, for ABET accredited engineering majors, at least 3/4 of the credits (3/8 + 1/4 + 1/8) are covered by some kind of requirement, leaving 1/4 as free electives. But at many schools, the subject requirements are greater and effectively mean that more than 3/4 of the courses are covered by some kind of requirement for many engineering majors, so that the amount of free electives is lower than 1/4.
Other science majors and other liberal arts majors commonly have about 1/4 of the credits in upper level course work for the major, with varying amounts of lower level course work as prerequisites. But there would also be general education requirements at most colleges.