How to display posttypes and taxonomy in standard posts, not in a separate label?

when you register a taxonomy, the second argument is the post type it is available for:

register_taxonomy( $taxonomy, $object_type, $args );

if you want the taxonomy to work for a custom type as well as standard posts, you can pass an array of types:

$object_type = array( 'your_custom_type', 'post' );

alternately, you can use register_taxonomy_for_object_type:

register_taxonomy_for_object_type( 'your_custom_taxonomy', 'post' );