The function checks what you want:
create or replace function json_has_id_and_name(val json)
returns boolean language sql as $$
select coalesce(
(
select array['id', 'name'] <@ array_agg(key)
from json_object_keys(val) key
),
false)
$$;
select json_has_id_and_name('{"id":1, "name":"abc"}'), json_has_id_and_name('{"id":1}');
json_has_id_and_name | json_has_id_and_name
----------------------+----------------------
t | f
(1 row)
You can use it in a check constraint, e.g.:
create table my_table (
id int primary key,
jdata json check (json_has_id_and_name(jdata))
);
insert into my_table values (1, '{"id":1}');
ERROR: new row for relation "my_table" violates check constraint "my_table_jdata_check"
DETAIL: Failing row contains (1, {"id":1}).