Joining to same table and updating column on row match for multiple columns
You can do it using a self-join and an aggregate function (string_agg) to aggregate the identifiers that are different from the current row as below.

The first and last lines initializes the status column to NULL and finalizes it to foo if it isn't assigned, respectively:

UPDATE table_1 t1 SET status = NULL;

UPDATE table_1 t1
SET status = (
  SELECT string_agg(t2.identifier,',') FROM table_1 t2
  WHERE t2.identifier != t1.identifier
    AND t1.col1 = t2.col1
    AND t1.col2 = t2.col2
    AND t1.col3 = t2.col3
    AND t1.col4 = t2.col4
);

UPDATE table_1 t1 SET status = 'foo' WHERE status  IS NULL;