Display post_meta-by_key on product catalogue

That’s because get_post_meta_by_key() isn’t a WordPress function.

What you’re likely looking for is get_post_meta():

add_action( 'woocommerce_after_shop_loop_item', 'display_author' );

function display_author() { 
  global $post;
  echo get_post_meta( $post->ID, 'author', true );
}

Not tested, because I don’t have a WooCommerce install handy (and WooCommerce specifics are off-topic here anyway) but this is pretty standard so it should just work for you.

The true as the third parameter is so that get_post_meta() returns a string, ready to echo directly out. Without true, it will return an array instead (see the docs for further info).