Postgres 9.4

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CREATE TABLE tbl(
    id int,
    column1 text,
    column2 text,
    column3 text
);

INSERT INTO tbl VALUES 
(1, NULL, 'A1', NULL),
(2, 'A2', NULL, NULL),
(3, NULL, NULL, NULL),
(4, 'A2', 'A3', 'A4'),
(5, NULL, NULL, 'A4'),
(6, 'A2', 'A3', NULL),
(7, 'A5', 'A3', NULL);
7 rows affected

CREATE OR REPLACE FUNCTION f_my_grouping()
  RETURNS SETOF int[] AS
$func$
DECLARE
   r  tbl; -- use table type as row variable
   r0 tbl;
   ids int[];
BEGIN
   FOR r IN
      SELECT * FROM tbl t ORDER BY t.id
   LOOP
      IF (r.column1, r.column2, r.column3) IS NULL THEN     -- all NULL
         RETURN NEXT ARRAY[r.id];  -- return and ignore

      ELSIF (r.column1 <> r0.column1 OR                     -- continue
             r.column2 <> r0.column2 OR
             r.column3 <> r0.column3) IS NOT TRUE  -- no mismatch
        AND (r.column1 =  r0.column1 OR
             r.column2 =  r0.column2 OR
             r.column3 =  r0.column3) THEN         -- 1+ match

         ids := ids || r.id;     -- add to array

         IF r0.column1 IS NULL AND r.column1 IS NOT NULL OR
            r0.column2 IS NULL AND r.column2 IS NOT NULL OR
            r0.column3 IS NULL AND r.column3 IS NOT NULL THEN

            SELECT INTO r0.column1, r0.column2, r0.column3 
                   COALESCE(r0.column1, r.column1)
                 , COALESCE(r0.column2, r.column2)
                 , COALESCE(r0.column3, r.column3);
         END IF;

      ELSE                                                  -- new grp
         IF r0 IS NULL THEN      -- skip 1st row
            -- do nothing
         ELSE
            RETURN NEXT ids;
         END IF;
         ids := ARRAY[r.id];     -- start new array
         r0  := r;               -- remember last row
      END IF;
   END LOOP;

   IF ids IS NOT NULL THEN  -- all NULL
      RETURN NEXT ids;  -- output last iteration
   END IF;
END
$func$  LANGUAGE plpgsql;
SELECT * FROM f_my_grouping();


f_my_grouping


{1}

{3}

{2,4,5,6}

{7}


-- If you require sorted output:
SELECT * FROM f_my_grouping() ORDER BY 1;


f_my_grouping


{1}

{2,4,5,6}

{3}

{7}


-- S-Man's clever pure SQL solution
WITH test_table as (
    SELECT *, 
    array_remove(ARRAY[column1,column2,column3], null) as arr, -- A
    cardinality(array_remove(ARRAY[column1,column2,column3], null))as arr_len
FROM tbl)

SELECT 
    s.array_agg as aggregates,                                 -- G
    MAX(tt.column1) as column1, 
    MAX(tt.column2) as column2, 
    MAX(tt.column3) as column3
FROM (

    SELECT array_agg(id) FROM                                  -- E
        (SELECT DISTINCT ON (t1.id)
        t1.id, CASE WHEN t1.arr_len >= t2.arr_len THEN t1.arr ELSE t2.arr END as arr  -- C
        FROM 
        test_table as t1 
        JOIN                                                   -- B
        test_table as t2
        ON t1.arr @> t2.arr AND COALESCE(t2.column1, t2.column2, t2.column3) IS NOT NULL
        OR t2.arr @> t1.arr AND COALESCE(t1.column1, t1.column2, t1.column3) IS NOT NULL

        ORDER BY t1.id, GREATEST(t1.arr_len, t2.arr_len) DESC -- D
        ) s
    GROUP BY arr  

    UNION

    SELECT 
        ARRAY[id] 
    FROM test_table tt 
    WHERE COALESCE(tt.column1, tt.column2, tt.column3) IS NULL) s -- F

JOIN test_table tt ON tt.id = ANY (s.array_agg)
GROUP BY s.array_agg;


aggregates
column1
column2
column3


{7}
A5
A3
null

{2,4,5,6}
A2
A3
A4

{1}
null
A1
null

{3}
null
null
null
aggregates	column1	column2	column3
{7}	A5	A3	null
{2,4,5,6}	A2	A3	A4
{1}	null	A1	null
{3}	null	null	null