Assigning the same custom meta box to multiple post types

Yes, you need to edit your video_add_box() function in order to make an array with the post type you need to show that box.

$postypes = array('type1', 'type2', 'type3');
foreach ( $postypes as $postype) {

    add_meta_box(
        'video_information',
        'Video Information',
        'video_information',
        $postype
    );
}

You can read more in codex: http://codex.wordpress.org/Function_Reference/add_meta_box

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