I see this FAQ in the Apple two-factor auth documentation:
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Will I need a trusted device dedicated to my Apple Developer account if I enable two-factor authentication?
No. You’ll need to use a trusted device to enable two-factor authentication for the first time. However, you can use the same trusted device for multiple Apple IDs that are enabled for two-factor authentication. Additionally, if you do not have access to your trusted device, you can get your verification code via SMS or phone call. When possible, you should use a trusted device to increase security and streamline the process.
[Emphasis mine.]
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As far as I can tell, this is simply not true. One must have a dedicated device because only one Apple ID can be logged into at any given moment and if one adds a second iCloud account, there is no way to adjust its security settings, as it's not the primary iCloud account on the device. So one must log out of the primary account and log in with the second account, but when I did so, I got the error: "Two-factor authentication is not available for your Apple ID at this time."
There is another FAQ that states:
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What if I’ve enabled two-step verification?
If your Apple ID has two-step verification enabled and two-factor authentication is available in your country or region, you will need to update to two-factor authentication for increased security.
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This pretty much rules out two-step verification as an alternative.
So far this has been a comedy of errors for me. I haven't found a clear discussion in the forums from others who may have this situation and what they've done to work around it. I'm currently unable to enable two-factor authentication for my developer account (and thus can't access my account dashboard), due to this limitation.
I would appreciate any help from anyone who has attempted to use multiple Apple IDs. Thanks!