wp_insert_post inside save_post adds wrong metadata to inserted post

The reason for that is the save_post loop. When you call wp_insert_post, it triggers save_post and thus inserts the student meta.

what you can do is, to check if the post type is correct while inserting the meta.

like:

add_action( 'save_post', 'save_student_meta', 10, 2 );

function save_student_meta( $post_id, $post ) {
    if ( 'student' !== $post->post_type ) {
        return;
    }
    // Code gathering data form $_POST