Count how many posts have a specified tag AND category

You could use WP_Query and specifically a tax_query:

    $args = array(
        'post_type' => 'post',
        'tax_query' => array(
            'relation' => 'AND',
            array(
                'taxonomy' => 'category',
                'field'    => 'slug',
                'terms'    => array( 'allow-list' ),
            ),
            array(
                'taxonomy' => 'post_tag',
                'field'    => 'slug',
                'terms'    => array( 'cats' ),
            ),
        ),
    );
    $the_query = new WP_Query( $args );
    if ( $the_query->have_posts() ) {
      $count = $the_query->found_posts;
      echo 'Post count: ' . $count;
     // echo '<ul>';

     // while ( $the_query->have_posts() ) {
      //  $the_query->the_post();
       // echo '<li>' . get_the_title() . '</li>';
     // }
    //  echo '</ul>';
   } else {
     echo 'No posts found';
   }
/* Restore original Post Data */
wp_reset_postdata();