Shortcode don’t work in custom option field, any solution?

If you have a shortcode string saved, try echoing it with do_shortcode function, which turns shortcodes into content.

$saved_shortcode = get_option( 'my_customizer', 'copyright' );
if ( $saved_shortcode && is_string( $saved_shortcode ) ) {
echo do_shortcode( $saved_shortcode );
}

More about the function on the developer handbook, https://developer.wordpress.org/reference/functions/do_shortcode/

EDIT

Another option is just echo the year directly on the footer template without first getting the shortcode string from the database and then rendering it.