Aggregate function to Overlapping dates in PostgreSQL

SOLUTION 1 : when all the date intervals within the same group are overlapping

SELECT id
     , group
     , min(start_date) AS start_date
     , max(end_date) AS end_date
     , Sum(quantity) AS quantity
  FROM mytable
GROUP BY id, group

SOLUTION 2 : when all the date intervals within the same group are not overlapping

Using a window function would be great but I don't see how to specify the window which corresponds to a subset of date ranges which overlap. So I come back to a basic solution based on a self-join which select for every row of table a the subset of rows of table b which overlap with that row :

SELECT DISTINCT ON (a.id, date_start, date_end)
       a.id
     , min(b.start_date) AS date_start
     , max(b.end_date) AS date_end
     , sum(b.quantity) AS quantity
  FROM mytable AS a
 INNER JOIN mytable AS b
    ON a.id = b.id
   AND a.start_date < b.end_date
   AND a.end_date > b.start_date
 GROUP BY a.id, a.start_date, a.end_date
 ORDER BY a.id, date_start

see the test result in dbfiddle

You may use the LAG function to check the consecutive overlapped date periods and create unique groups for them as the following:

WITH create_groups AS
(
  SELECT *, SUM(chk) OVER (PARTITION BY id, group_ ORDER BY start_date) grp
  FROM
   (
     SELECT *,
        CASE 
         WHEN start_date <= LAG(end_date, 1, end_date) OVER (PARTITION BY id, group_ ORDER BY start_date) AND
              start_date >=LAG(start_date, 1, start_date) OVER (PARTITION BY id, group_ ORDER BY start_date) 
         THEN 0 ELSE 1
        END AS chk
     FROM table_name
  ) T
)
SELECT id, group_, 
       MIN(start_date) start_date,
       MAX(end_date) start_date,
       SUM(quantity) quantity
FROM create_groups
GROUP BY id, group_, grp
ORDER BY id, group_, grp

See a demo.