Students cannot set the school that shows up on their Profile. A student's school affiliation is added automatically to that of their teacher's when they join a class.
When a student joins a class whose owner is part of a different school than that listed on their Profile, the student will be given an opportunity to update their school affiliation.
Note: Teachers can also only have one school listed on their Profile. If a teacher works at two schools, then they should pick one to associate with their Profile and add their other school's information to the About Me section of their Profile.