Postgres 10

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create table t  as select * from ( VALUES(1, '[{ "menu_id":"0", "menu_name":"name 1"},
    { "menu_id":"1", "menu_name":"name 2"},
    { "menu_id":"2", "menu_name":"name 3"}]' :: text) ) as s(client_id,col);

SELECT 1

select * from t

client_id
col

1
[{ "menu_id":"0", "menu_name":"name 1"},
    { "menu_id":"1", "menu_name":"name 2"},
    { "menu_id":"2", "menu_name":"name 3"}]

SELECT 1

update "t" set "col"= jsonb_set("cols":: JSONB,('{'|| elem_index || ',"' ||'menu_name"}')::text[],'"name updated"'::jsonb,false)
from (select pos- 1 as elem_index,"col" as "cols","client_id" as "colId"
	from  "t",jsonb_array_elements("t"."col" :: JSONB) with ordinality arr(elem, pos)    
where elem->>'menu_id' = '2') as "tble"     
    where  client_id =1

UPDATE 1

select * from t

client_id
col

1
[{"menu_id": "0", "menu_name": "name 1"}, {"menu_id": "1", "menu_name": "name 2"}, {"menu_id": "2", "menu_name": "name updated"}]

SELECT 1