Edited code and got it working. Thank you, @nico_haase!

$instituteEntities = [];

        for ($i = 0; $i < count($instituteList); $i++) {
            $institute = new Institute();
            $institute->setName($instituteList[$i]);
            $instituteEntities[] = $institute;
            $manager->persist($institute);
        }

        for ($i = 0; $i < count($departmentList); $i++) {
            $department = new Department();
            $department->setName($departmentList[$i]);
            $department->setInstitute($instituteEntities[rand(0, count($instituteEntities) - 1)]);
            $manager->persist($department);
        }