Better database for "keep always the 5 latest entries per ID and delete older"?

One option I can think of is to delete the oldest row as soon as you insert a new one. That will only require a very quick lookup limited to at most 6 rows rather then going through all rows at a time.

To do that efficiently you need a unique key on the table:

create table history 
(
  id          serial primary key, -- to make a lookup on a single row efficient
  fooid       text not null,
  target      text not null,
  updated_at  timestamp not null default now()
);
-- to make finding the oldest row for one fooid efficient
create index on history(fooid, updated_at);

Then create a trigger that only keeps the 5 most recent rows for a fooid:

create or replace function remove_last() 
  returns trigger
as
$$
begin
  with ranked as (
    select id, row_number() over (partition by fooid order by updated_at) as rn
    from history
    where id <> new.id
      and fooid = new.fooid 
  )
  delete from history
  where id in (select id 
               from ranked 
               where rn >= 5);
  return new;
end;
$$
language plpgsql;

create trigger remove_last_trigger 
  after insert on history
  for each row execute procedure remove_last();

On my laptop, a simple test setup with 5 million rows (1 million different fooid values) showed a trigger overhead that was less then a millisecond:

insert into history (fooid, target) values ('1', 'new stuff');

QUERY PLAN                                                                                          
----------------------------------------------------------------------------------------------------
Insert on stuff.history  (cost=0.00..0.01 rows=1 width=76) (actual time=0.062..0.062 rows=0 loops=1)
  Buffers: shared hit=8                                                                             
  ->  Result  (cost=0.00..0.01 rows=1 width=76) (actual time=0.017..0.017 rows=1 loops=1)           
        Output: nextval('history_id_seq'::regclass), '1'::text, 'new stuff'::text, now()            
        Buffers: shared hit=1                                                                       
Planning time: 0.024 ms                                                                             
Trigger remove_last_trigger: time=0.438 calls=1
Execution time: 0.524 ms

How much overhead that is in your system and whether or not that is acceptable for you I don't know, but "thousand changes per day" doesn't sound like a very busy system.