Assuming that you want to update the new column with the list of descriptions of all IPs of each record_no , then, starting from your existing query, you would need to correlate the subquery to the outer query (and drop the having clause):

update r
set concatenated_ip = (
    select string_agg(final_ip_description, ' | ') 
        within group (order by final_ip_description)
    from ip_tiebreaker_ref_table r1 
    where r1.record_no = r.record_no  -- correlation
)
from ip_tiebreaker_ref_table r

Note that this assigns the same value to all rows that have the same record_no ( within group is there to make the sort predictable) - which one might consider data duplication - and it is hard to maintain such information up to date when the data changes later on.

Another approach would be to use a view, which gives you an always up-to-date perspective at your data for 0 maintenance costs:

create view v_ip_tiebreaker_ref_table as
select r.*, r1.*
from ip_tiebreaker_ref_table r
cross apply (
    select string_agg(final_ip_description, ' | ') 
        within group (order by final_ip_description) as concatenated_ip
    from ip_tiebreaker_ref_table r1 
    where r1.record_no = r.record_no  -- correlation
) r1

The problem with your query is that your subquery is not correctly linked to the single records you're trying to update, as long as your subqueries returns multiple rows.

In order to fix this problem, you can try using the FROM... WHERE clauses featured in SQL Server UPDATE statements.

UPDATE [dm_na_fin_da].[ip_tiebreaker_ref_table] 
SET Concatenated_IP = cte.descr
FROM (SELECT record_no, STRING_AGG(final_ip_description, ' | ') AS descr
      FROM [dm_na_fin_da].[updated_ip_dataset] 
      GROUP BY record_no) cte
WHERE [dm_na_fin_da].[ip_tiebreaker_ref_table].record_no = cte.record_no

You can update using INNER JOIN as follows :

UPDATE [ip_tiebreaker_ref_table]
SET Concatenated_IP = s.Concatenated_IP
FROM [ip_tiebreaker_ref_table] it
INNER JOIN (
  select uid.record_no, string_agg(final_ip_description, ' | ') as Concatenated_IP
  from [updated_ip_dataset] uid
  group by uid.record_no
) as s on it.record_no = s.record_no

Demo here