Remove “Get Shortlink” button in admin of custom post type

If you filter pre_get_shortlink and return anything but false WordPress will not create a shortlink with its own logic. If your return value is empty, the shortlink UI will not be printed.

Combining both leads us to:

add_filter( 'pre_get_shortlink', '__return_empty_string' );

If you want to restrict the filter to a specific post type, check the second parameter:

add_filter( 'pre_get_shortlink', function( $false, $post_id ) {
    return 'page' === get_post_type( $post_id ) ? '' : $false;
}, 10, 2 );