Actually, this generalization may be less true than commonly assumed on these forums. Intended major may affect what division one applies to (e.g. College of Arts and Science versus College of Engineering), which may result in different admission selectivity. Even some of the most selective private schools (Penn, Columbia, Cornell) have admission by division or major, as do many public universities (including flagships).
Also be aware of schools where you are admitted as an undeclared student, but your intended major requires a high college GPA or competitive admission to declare later.