That's not what's happening at all. They are using a service called Plaid for bank authentication. It's used a lot of places. And you are doing ACH transfers; not bank wires.
Also from https://plaid.com/legal/longtail/, "You acknowledge that in accessing your data and information through the Service, your provider account access number(s), password(s), security question(s) and answer(s), account number(s), login information, and any other security or access information, and the actual data in your account(s) with such provider(s) such as bank and other account balances, credit card charges, debits and deposits (collectively, "Provider Account Data"), may be collected and stored in the Service."