I would log in to the main page where we can view government services. Then I would click through to the ATO (our IRS) to see some statements or whatever. I would then close the window and log out in MyGov then log back in with my wife's account. Click through to the ATO would show my details instead of hers. IIRC logging out wouldn't work either since that would mess up the session, I'd have to come back later.
Couldn't find anywhere to report this so I ignored it. I think they fixed it now.