Synchronize custom post type tags with WordPress default post type tags

register_taxonomy_for_object_type() will do what you want as long as the taxonomy and object (post_type) are already registered elsewhere. Very useful when dealing with a plugin’s CPTs you need to add a taxonomy to.

add_action( 'init', 'add_tag_tax_to_posts');

function add_tag_tax_to_posts(){

  register_taxonomy_for_object_type( 'SLUG_OF_PLUGIN_TAG_TAX_HERE', 'post' );

}

Or to do the reverse, and add the default tags (post_tag) to the CPTs:

register_taxonomy_for_object_type( 'post_tag', 'SLUG_OF_PLUGIN_CPT_HERE' );