Associate Taxonomies Across Post Types

Create one taxonomy and share it across all post types. You won’t be able to have the same slug across multiple taxonomies anyway, they have to be unique due to the way they’re stored in the database.

You can use the built in tags taxonomy with your custom post type by setting

'taxonomy' => array('post_tag')

when you register your custom post type, or you can add a custom taxonomy to regular posts (as well as your custom types) when you register your taxonomy by passing an array for $object_type:

array('post', 'post_type_author', 'post_type_book')