As pointed out, if you want to communicate with two different databases from a client program, you need two connections thus two connection strings.
However, since you're moving data from one database to another, I would consider not transferring the data via a client program. When you use a client program you first need to read the data into the program, perhaps process it and then write it to the other database.
Instead, I would suggest using a
linked server. A linked server is a server side object that can be used to communicate directly from a database to another. I believe that this approach would be more straightforward and also more efficient. In the best case you could perhaps use a simple
INSERT INTO ... SELECT FROM
structure to transfer the data.
For more information, have a look at
Linked Servers (Database Engine) | Microsoft Docs[
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