Synchronize Custom post type tags to WordPress default posts tags

You need to register the post_tag taxonomy for the post type.

For a quick, after the fact, way to do this (meaning the CPT has already been registered, etc.), you can use register_taxonomy_for_object_type().

A workable example is below. Edit the value for $object_type, and place the example below in the functions.php of your child theme:

  add_action( 'init', 'my_cpt_has_tags_now' );
  function my_cpt_has_tags_now() {
       $taxonomy    = "post_tag";
       $object_type = "name_of_your_cpt";

       register_taxonomy_for_object_type( $taxonomy, $object_type );
  }