Postgres 11

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WITH session_log AS (
  SELECT 'ABC' as site_id, 1234 user_id, 12 session_id, '2020-02-10 00:29:59.376000 UTC' start_time, '2020-02-10 01:13:02.817000 UTC' end_time UNION ALL
  SELECT 'ABC' as site_id, 1234 user_id, 13 session_id, '2020-02-10 02:41:56.330000 UTC' start_time, '2020-02-10 02:41:56.389999 UTC' end_time UNION ALL
  SELECT 'ABC' as site_id, 1234 user_id, 14 session_id, '2020-02-10 04:24:46.649999 UTC' start_time, '2020-02-10 05:14:08.243000 UTC' end_time UNION ALL
  SELECT 'ABC' as site_id, 1234 user_id, 15 session_id, '2020-02-10 04:59:21.356999 UTC' start_time, '2020-02-10 15:57:11.501000 UTC' end_time UNION ALL
  SELECT 'ABC' as site_id, 6789 user_id, 25 session_id, '2020-02-10 02:15:38.560000 UTC' start_time, '2020-02-10 02:56:38.784500 UTC' end_time UNION ALL
  SELECT 'ABC' as site_id, 6789 user_id, 26 session_id, '2020-02-10 04:59:21.356999 UTC' start_time, '2020-02-10 15:57:11.501000 UTC' end_time
)
select
 (select count(*) 
   from session_log as xtra_sess
   where logins.site_id = xtra_sess.site_id
   and logins.user_id = xtra_sess.user_id
   and logins.session_id <> xtra_sess.session_id
   and (xtra_sess.start_time between logins.start_time and logins.end_time
     or xtra_sess.end_time between logins.start_time and logins.end_time
     )
  ) xtras_sess
  , *
from session_log as logins

xtras_sess	site_id	user_id	session_id	start_time	end_time
0	ABC	1234	12	2020-02-10 00:29:59.376000 UTC	2020-02-10 01:13:02.817000 UTC
0	ABC	1234	13	2020-02-10 02:41:56.330000 UTC	2020-02-10 02:41:56.389999 UTC
1	ABC	1234	14	2020-02-10 04:24:46.649999 UTC	2020-02-10 05:14:08.243000 UTC
1	ABC	1234	15	2020-02-10 04:59:21.356999 UTC	2020-02-10 15:57:11.501000 UTC
0	ABC	6789	25	2020-02-10 02:15:38.560000 UTC	2020-02-10 02:56:38.784500 UTC
0	ABC	6789	26	2020-02-10 04:59:21.356999 UTC	2020-02-10 15:57:11.501000 UTC

WITH session_log AS (
  SELECT 'ABC' as site_id, 1234 user_id, 12 session_id, '2020-02-10 00:29:59.376000 UTC' start_time, '2020-02-10 01:13:02.817000 UTC' end_time UNION ALL
  SELECT 'ABC' as site_id, 1234 user_id, 13 session_id, '2020-02-10 02:41:56.330000 UTC' start_time, '2020-02-10 02:41:56.389999 UTC' end_time UNION ALL
  SELECT 'ABC' as site_id, 1234 user_id, 14 session_id, '2020-02-10 04:24:46.649999 UTC' start_time, '2020-02-10 05:14:08.243000 UTC' end_time UNION ALL
  SELECT 'ABC' as site_id, 1234 user_id, 15 session_id, '2020-02-10 04:59:21.356999 UTC' start_time, '2020-02-10 15:57:11.501000 UTC' end_time UNION ALL
  SELECT 'ABC' as site_id, 6789 user_id, 25 session_id, '2020-02-10 02:15:38.560000 UTC' start_time, '2020-02-10 02:56:38.784500 UTC' end_time UNION ALL
  SELECT 'ABC' as site_id, 6789 user_id, 26 session_id, '2020-02-10 04:59:21.356999 UTC' start_time, '2020-02-10 15:57:11.501000 UTC' end_time
)
select
  count(distinct xtra_sess.site_id) as xtra_sess
, logins.site_id
, logins.user_id
, logins.session_id
, logins.start_time
, logins.end_time
from session_log as logins
left join session_log as xtra_sess
   on  logins.site_id = xtra_sess.site_id
   and logins.user_id = xtra_sess.user_id
   and logins.session_id <> xtra_sess.session_id
   and (xtra_sess.start_time between logins.start_time and logins.end_time
     or xtra_sess.end_time between logins.start_time and logins.end_time
     )
group by
  logins.site_id
, logins.user_id
, logins.session_id
, logins.start_time
, logins.end_time

xtra_sess	site_id	user_id	session_id	start_time	end_time
0	ABC	1234	12	2020-02-10 00:29:59.376000 UTC	2020-02-10 01:13:02.817000 UTC
0	ABC	1234	13	2020-02-10 02:41:56.330000 UTC	2020-02-10 02:41:56.389999 UTC
1	ABC	1234	14	2020-02-10 04:24:46.649999 UTC	2020-02-10 05:14:08.243000 UTC
1	ABC	1234	15	2020-02-10 04:59:21.356999 UTC	2020-02-10 15:57:11.501000 UTC
0	ABC	6789	25	2020-02-10 02:15:38.560000 UTC	2020-02-10 02:56:38.784500 UTC
0	ABC	6789	26	2020-02-10 04:59:21.356999 UTC	2020-02-10 15:57:11.501000 UTC