You need to supply the post id to update the field on. During an AJAX request WP and ACF will not know what post you’re trying to update unless you tell it. The thread you linked to has the post id in a hidden field as well. You need to supply the post ID when updating the field
update_field($field_name, $value, $post_id);
see https://www.advancedcustomfields.com/resources/update_field/
