MySQL 8.0

CREATE TABLE tournaments (id INT PRIMARY KEY, game VARCHAR(64));
INSERT INTO tournaments (`id`, `game`) VALUES ('1', 'EXAMPLE'), ('2', 'TETRIS'), ('3', 'MARIO'), ('4', 'EXAMPLE'), ('5', 'TETRIS'), ('6', 'MARIO');
CREATE TABLE matches (id INT PRIMARY KEY, tournament_id INT, date_start DATETIME, date_finished DATETIME);
INSERT INTO matches (`id`, `tournament_id`, `date_start`, `date_finished`) VALUES ('1', '1', '2024-01-18 01:10:00', '2024-01-18 01:12:00'), ('2', '2', '2024-01-18 01:10:00', '2024-01-18 01:18:00'), ('3', '3', '2024-01-18 01:10:00', '2024-01-18 01:16:00'), ('4', '3', '2024-01-18 01:10:00', '2024-01-18 01:22:00'), ('5', '2', '2024-01-18 01:10:00', '2024-01-18 01:31:00'), ('6', '1', '2024-01-18 01:10:00', '2024-01-18 01:19:00'), ('7', '2', '2024-01-18 01:10:00', '2024-01-18 01:45:00'), ('8', '4', '2024-01-18 01:10:00', '2024-01-18 01:28:00'), ('9', '4', '2024-01-18 01:10:00', '2024-01-18 01:54:00'), ('10', '5', '2024-01-18 01:10:00', '2024-01-18 01:12:00'), ('11', '6', '2024-01-18 01:10:00', '2024-01-18 01:18:00'), ('12', '7', '2024-01-18 01:10:00', '2024-01-18 01:16:00'), ('13', '7', '2024-01-18 01:10:00', '2024-01-18 01:22:00'), ('14', '6', '2024-01-18 01:10:00', '2024-01-18 01:31:00'), ('15', '5', '2024-01-18 01:10:00', '2024-01-18 01:19:00'), ('16', '6', '2024-01-18 01:10:00', '2024-01-18 01:45:00'), ('17', '6', '2024-01-18 01:10:00', '2024-01-18 01:28:00'), ('18', '5', '2024-01-18 01:10:00', '2024-01-18 01:54:00');

Records: 6  Duplicates: 0  Warnings: 0

Records: 18  Duplicates: 0  Warnings: 0

Query was empty

SELECT 
 t.game
, avg(`val`) as `median`
FROM
(
    SELECT
        `game`,
        `val`,
        (SELECT count(*) 
           FROM `matches` `t2` JOIN tournaments ta ON ta.id = t2.tournament_id
          WHERE `ta`.`game` = `t3`.`game` AND ta.id NOT IN (5, 6)) as `ct`,
        `seq`,
        (SELECT count(*) FROM `matches` `t2` JOIN tournaments ta ON ta.id = t2.tournament_id
         WHERE `ta`.`game` < `t3`.`game`  AND ta.id NOT IN (5, 6)) as `delta`

FROM
        (SELECT `game`, `val`, @rownum := @rownum + 1 as `seq`
        FROM (SELECT `game`, (TIMESTAMPDIFF(minute, date_start, date_finished)) `val` 
               FROM `matches` m JOIN tournaments ta ON ta.id = m.tournament_id
              WHERE ta.id NOT IN (5, 6)
               ORDER BY `game`, `val`) as `t2`
        CROSS JOIN (SELECT @rownum := 0) as `x`
        ORDER BY `game`, `seq`) as `t3`
    HAVING
        (`ct`%2 = 0 and `seq`-`delta` between floor((`ct`+1)/2) and floor((`ct`+1)/2) +1)
        or (`ct`%2 <> 0 and `seq`-`delta` = (`ct`+1)/2)
) as `t`

GROUP BY t.game
ORDER BY t.game;

game	median
EXAMPLE	13.5000
MARIO	9.0000
TETRIS	21.0000