Postgres 11

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CREATE TABLE users ( attrs JSON NOT NULL );

INSERT INTO users VALUES 
('{"info": { "x": "abc" ,"date_of_birth" : "02:21:2020" }}'),
('{"info": { "y": "def" ,"date_of_birth" : "2000-11-03" }}');
2 rows affected

ALTER TABLE users ALTER COLUMN attrs 
  SET DATA TYPE JSONB USING attrs::JSONB
SELECT '"'||TO_DATE(attrs::JSON->'info'->>'date_of_birth', 'mm:dd:yyyy')::TEXT||'"'
  FROM users
 WHERE POSITION( ':' IN attrs::JSON->'info'->>'date_of_birth'::TEXT ) > 0


?column?


"2020-02-21"


UPDATE users
   SET attrs = JSONB_SET( attrs, '{info,date_of_birth}', 
                          ('"'||TO_DATE(
                                  attrs::JSONB->'info'->>'date_of_birth', 'mm:dd:yyyy'
                                  )::TEXT||'"')::JSONB 
                        )
 WHERE POSITION( ':' IN attrs::JSON->'info'->>'date_of_birth'::TEXT ) > 0                        
1 rows affected

SELECT *
  FROM users


attrs


{"info": {"y": "def", "date_of_birth": "2000-11-03"}}

{"info": {"x": "abc", "date_of_birth": "2020-02-21"}}