I called Apple Support and the solution was to turn off Personal Requests for both sets of Pods, then turn Personal Requests back on again and then restart both pairs of HomePods. This fixed the problem and I can now use the HomePods to place calls. During the call, it came up that my wife and I share the same AppleID to allow us to share the iCloud Photo Library. He said that Apple does not recommend sharing the same ID and this is why I am unable to invite my wife to join My Home in the Home App. Not sure of the ramifications of this but it may explain why I haven't been able to get Intercom working and why Siri doesn't recognize our different voices. Ironic, because it was Apple that recommended sharing the same ID to facilitate sharing photos but that was several years ago. I'll have to think about how to best share photos if we aren't on the same ID.
Edit: Turns out Intercom does work now since resetting the HomePods. The function I cannot use is Multiuser because we share the same AppleID.