It seems that you are using a username that is linked with the Marketing Cloud Log in or you are using Marketing Cloud ID (email) to log in for APIs or BAAAM.
Please log onto AAM UI and create a new user and use an email address that is not a Marketing Cloud ID. Then, check by logging onto http://bank.demdex.com and see if you are able to log in with username and password only. If you are able to log in without being redirected to Marketing Cloud sign in, then you will be able to use that username and password in APIs.
For further assistance on APIs or BAAAM, you can contact AAM support.
With the recent Experience Cloud level user management changes. Enterprises are required to have an API account / user to access APIs which is not tied to Experience cloud. We would recommend to contact your consultant or AAM support (firstname.lastname@example.org)