I have two separate databases. I am trying to update a column in one database to the values of a column from the other database:
UPDATE customer
SET customer
This error means that the SELECT store_key FROM store query has returned two or more rows in the SERVER1 database. If you would like to update all customers, use a join instead of a scalar = operator. You need a condition to "connect" customers to store items in order to do that.
If you wish to update all customer_ids to the same store_key, you need to supply a WHERE clause to the remotely executed SELECT so that the query returns a single row.