MySQL 5.7

By using db<>fiddle, you agree to license everything you submit by Creative Commons CC0.
select version();

CREATE TABLE `table_a` (
  `id` int(11) NOT NULL,
  `references` json NULL
);

CREATE TABLE `table_b` (
  `id` int(11) NOT NULL,
  `name` text NULL
);

INSERT INTO `table_a` (`id`, `references`) VALUES
(1, '\"[1,3]\"'),
(2, '\"[2,3]\"');

INSERT INTO `table_b` (`id`, `name`) VALUES
(1, 'item_1'),
(2, 'item_2'),
(3, 'item_3');

SELECT * from table_a;
SELECT * from table_b;

select `references` from table_a where id=1;

SELECT JSON_SEARCH((select `references` from table_a where id=1), 'one', '3');

version()

5.7.29

id
references

1
"[1,3]"

2
"[2,3]"

id
name

1
item_1

2
item_2

3
item_3

references

"[1,3]"

JSON_SEARCH((select `references` from table_a where id=1), 'one', '3')

null

SELECT JSON_SEARCH('[1,3]', 'one', 1);
SELECT JSON_SEARCH('"[1,3]"', 'one', 1);
SELECT JSON_SEARCH('["1","3"]', 'one', 1);
SELECT JSON_SEARCH('"["1","3"]"', 'one', 1);

JSON_SEARCH('[1,3]', 'one', 1)

null

JSON_SEARCH('"[1,3]"', 'one', 1)

null

JSON_SEARCH('["1","3"]', 'one', 1)

"$[0]"

SELECT *, 
       JSON_VALID(table_a.references) checking,
       REPLACE(REPLACE(REPLACE(REPLACE(table_a.references, '"', ''), '[', '["'), ']', '"]'), ',', '","') valid_JSON
FROM table_a;

id
references
checking
valid_JSON

1
"[1,3]"
1
["1","3"]

2
"[2,3]"
1
["2","3"]

SELECT table_b.*
FROM table_a, table_b
WHERE table_a.id = 1
  AND JSON_SEARCH(REPLACE(REPLACE(REPLACE(REPLACE(table_a.references, '"', ''), '[', '["'), ']', '"]'), ',', '","'), 'one', table_b.id) IS NOT NULL

id
name

1
item_1

3
item_3