mysql если совпадает весь список

WITH
cte1 AS ( SELECT post_id, GROUP_CONCAT(DISTINCT category ORDER BY category) cat_list
          FROM category
          GROUP BY post_id
          HAVING SUM(category = 'car') ),
cte2 AS ( SELECT cat_list
          FROM cte1
          GROUP BY cat_list
          HAVING COUNT(cat_list) > 1 )
SELECT post_id, cat_list
FROM cte1
JOIN cte2 USING (cat_list)
ORDER BY 2,1;

fiddle

Для версии 5.x это будет выглядеть так:

SELECT post_id, cat_list
FROM ( SELECT post_id, GROUP_CONCAT(DISTINCT category ORDER BY category) cat_list
       FROM category
       GROUP BY post_id
       HAVING SUM(category = 'car') ) cte1
JOIN ( SELECT cat_list
       FROM ( SELECT post_id, GROUP_CONCAT(DISTINCT category ORDER BY category) cat_list
              FROM category
              GROUP BY post_id
              HAVING SUM(category = 'car') ) cte1
       GROUP BY cat_list
       HAVING COUNT(cat_list) > 1 ) cte2 USING (cat_list)
ORDER BY 2,1;

fiddle

Если нужно - добавьте в источник данных внешнего запроса и таблицу posts, соответствующим образом откорректировав список полей вывода.

Если необходимо получить только первое совпадение, можно использовать LIMIT

SELECT posts.*  
FROM posts 
JOIN categories ON categories.post_id = posts.id 
WHERE categories.category IN ("car","bed") 
LIMIT 1