There are 2 types of SQL Server database login. There's Windows Authentication and there is SQL Server Authentication. All logins to the database are set up by the DBA. In your case it sounds like your DBA has set up a Windows Authentication login for you which use your Window login info.
Yes, a user can log into Toad using something other then their computer login info. But this must be set up by the DBA for the database and is not controlled by Toad. You would need to ask your DBA for a new user name and password set up as SQL Server Authentication.
In the case where you would like to use another Windows Auth user to connect to the database you would need to use the Windows run as command and specify the other Windows user. In this case, Toad will pick up whatever is registered in the Windows and use this as the Windows Auth, accepting another Windows user login to the database.