Washington is not the only such school. Purdue and Minnesota also require engineering students to apply to their majors after completing the frosh year prerequisites.
However, the stress level at Washington for those who were not directly admitted to their desired engineering majors as frosh is likely much higher, due to the high GPA standards and holistic admission review for at least some engineering majors. Purdue and Minnesota students are assured of admission to their desired engineering majors with a 3.2 GPA, and the less popular majors likely have space available for most students with lower GPAs. In contrast, a 3.2 GPA at Washington probably means rejection to most engineering majors other than industrial, according to https://www.engr.washington.edu/current/admissions/admitstats .
The plus side of Washington is that a small number of frosh do get direct admission to engineering majors. A student who does get such a direct admission to his/her desired engineering major and is unlikely to change majors into another highly competitive major may find Washington to be a very desirable school to attend (assuming other constraints like cost are not a problem). But a student who does not receive direct admission to a highly competitive major would be taking a big risk by attending Washington, and should make backup plans to transfer elsewhere or finish in a less competitive major in the event that s/he is not admitted to the desired major.