Technically, to repair your statement, you can add LIMIT 1 to the subquery to ensure that at most 1 row is returned. That would remove the error, your code would still be nonsense.
... 'SELECT store_key FROM store LIMIT 1' ...
Practically, you want to match rows somehow instead of picking an arbitrary row from the remote table store to update every row of your local table customer.
Your rudimentary question doesn’t provide enough details, so I am assuming a text column match_name in both tables (and UNIQUE in store) for the sake of this example:
... 'SELECT store_key FROM store
WHERE match_name = ' || quote_literal(customer.match_name) ...
But that’s an extremely expensive way of doing things.
Ideally, you completely rewrite the statement.
UPDATE customer c
SET customer_id = s.store_key
FROM dblink('port=5432, dbname=SERVER1 user=postgres password=309245'
, 'SELECT match_name, store_key FROM store')
AS s(match_name text, store_key integer)
WHERE c.match_name = s.match_name
AND c.customer_id IS DISTINCT FROM s.store_key;
This remedies a number of problems in your original statement.
Obviously, the basic problem leading to your error is fixed.
It’s typically better to join in additional relations in the FROM clause of an UPDATE statement than to run correlated subqueries for every individual row.
When using dblink, the above becomes a thousand times more important. You do not want to call dblink() for every single row, that’s extremely expensive. Call it once to retrieve all rows you need.
With correlated subqueries, if no row is found in the subquery, the column gets updated to NULL, which is almost always not what you want. In my updated query, the row only gets updated if a matching row is found. Else, the row is not touched.
Normally, you wouldn’t want to update rows, when nothing actually changes. That’s expensively doing nothing (but still produces dead rows). The last expression in the WHERE clause prevents such empty updates:
AND c.customer_id IS DISTINCT FROM sub.store_key
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