UPDATE with join condition on matching words in columns of another table
The key here is that Column_1, represents three possible values for the JOIN. So what you want to use is string_to_array() (so long as those values are comma-separated and can not themselves include a comma).

Run this query,

SELECT id_number, string_to_array(column_1, ',') AS column_1
FROM table_a;
 id_number |       column_1        
-----------+-----------------------
       123 | {foo,bar,baz}
       456 | {qux,quux,quuz}
       789 | {corge,grault,garply}
       101 | {qux,bar,grault}


Now, we can run our UPDATE using = ANY(),

UPDATE table_b
SET id_number = A.id_number
FROM (
  SELECT id_number, string_to_array(column_1, ',') AS column_1
  FROM table_a
) AS A
WHERE table_b.column_1 = ANY(A.column_1)
  AND table_b.column_2 = ANY(A.column_1);


You can alternatively use &lt;@

WHERE ARRAY[table_b.column_1, table_b.column_2] &lt;@ A.column_1;


That even makes it a bit more compact..

UPDATE table_b
  SET id_number = A.id_number
FROM table_a AS A                              
  WHERE ARRAY[table_b.column_1, table_b.column_2] &lt;@ string_to_array(A.column_1, ',');