Ehm, they do here, both the Live and the Exchange accounts, and it’s possible to turn off 2FA in the tenant if so desired.
Something did happen in the MS servers today which apparently kicked all the users out.
The issue is when the authentication fails in EMC, and one has previously logged into a Live account in the browser and clicked the “Keep me logged in” checkbox (or yes to “Stay signed in”), leaving it go for some time.
Attempting to log in on the browser may initially fail authentication, as MS security does not inform you that you are already logged in.
In that case try the “use another account” option in the login prompt dialog when it then (hopefully) defaults to the same Live account as before, and will log you in successfully, and presto, no more EMC errors.