Adding a tag_ID column into Categories inside the admin dashboard?

You can do it like so:

1. Use the manage_<screen ID>_columns filter to add the “ID” column.

   Example for the default category taxonomy where <screen ID> is edit-category — for other taxonomies, it would be edit-<taxonomy> like edit-post_tag:

   add_filter( 'manage_edit-category_columns', 'my_manage_edit_category_columns' );
   function my_manage_edit_category_columns( $columns ) {
       // Add the "ID" column.
       $columns['term_id'] = 'ID';
       // .. add other columns, if you want to.
   
       return $columns;
   }
   

2. Use the manage_<screen ID>_sortable_columns filter to make the “ID” column be sortable.

   Example for the default category taxonomy (the screen ID format is the same as above):

   add_filter( 'manage_edit-category_sortable_columns', 'my_manage_edit_category_sortable_columns' );
   function my_manage_edit_category_sortable_columns( $sortable_columns ) {
       // Add the "ID" column.
       $sortable_columns['term_id'] = 'term_id';
       // .. add other columns, if you want to.
   
       return $sortable_columns;
   }
   

3. Use the manage_<taxonomy>_custom_column filter to display the actual term ID in the “ID” column.

   Example for the default category taxonomy:

   add_filter( 'manage_category_custom_column', 'my_manage_category_custom_column', 10, 3 );
   function my_manage_category_custom_column( $content, $column_name, $term_id ) {
       switch ( $column_name ) {
           case 'term_id':
               $content = $term_id;
               break;
       }
   
       return $content;
   }
   

That’s all, but for sorting by other fields than the term_id, you would need to manually filter the terms query args, e.g. using the get_terms_args filter or the parse_term_query action.