It looks there are no good built-in ways of doing this and there are still no good answers on this topic, so I thought I'd add my quick & dirty solution. If you execute the below code it will create a function called MY_JSON_INTERSECT that will output results exactly as the original poster specified. Make sure you've looked this over and are ok with creating a new function before trusting my code:
delimiter $$
CREATE FUNCTION `MY_JSON_INTERSECT`(Array1 VARCHAR(1024), Array2 VARCHAR(1024)) RETURNS varchar(1024)
BEGIN
DECLARE x int;
DECLARE val, output varchar(1024);
SET output = '[]';
SET x = 0;
IF JSON_LENGTH(Array2) < JSON_LENGTH(Array1) THEN
SET val = Array2;
SET Array2 = Array1;
SET Array1 = val;
END IF;
WHILE x < JSON_LENGTH(Array1) DO
SET val = JSON_EXTRACT(Array1, CONCAT('$[',x,']'));
IF JSON_CONTAINS(Array2,val) THEN
SET output = JSON_MERGE(output,val);
END IF;
SET x = x + 1;
END WHILE;
IF JSON_LENGTH(output) = 0 THEN
RETURN NULL;
ELSE
RETURN output;
END IF;
END$$
You can then call the function like this:
SELECT MY_JSON_INTERSECT('[1,2,3,4,5,6,7,8]','[0,3,5,7,9]');
Outputs:
[3,5,7]
This isn't beautiful or efficient, but it's something that works... Hopefully better answers will come soon.