Yes and no.

They sort of have shareholders in their Board of Trustees whose goals are to see the school grow. They also have shareholders in the form of their upper administration whose goals are usually to increase their own salaries. (Private universities are their own thing and have actual investors and shareholders and don't really count imo)

Really the problem with (public) universities though is the lack of any market pressure with no regulation to back it up. Student loans are federally guaranteed to never default, which means universities are guaranteed payment as long as students sign up. Incoming freshmen do not have the initial barrier to entry with loans, so their demand has zero response to price increases.Thus universities have zero market incentive to actually reduce any costs or optimize efficiency as a result, because fuck them kids, they're gonna get paid no matter what. It creates a poor feedback loop where the school's budget balloons, tuition is raised to make up the difference, students take out more loans, rinse, repeat.