At the moment drive (in contrast to mail) does not allow to quickly pull up the side bar, press the user bubble and select another account. It is always required to sign out of drive and sign in with the other account.

In mail, where I use the same setup, 1) my account and 2) my family account and with just a few tips I can switch between the inboxes. Drive on the other hand, does not allow this, there is just a single account that can be signed in.