Stumbled across a fix. Had this issue since HomePods 11.4 update with a guest to my home being able to accept on one device and it not syncing to their other (iPhone vs. iPad Pro and vise versa). Another had zero issues and it worked as it should.
Accidentally deleted the Home messing around with the wrong iPad whilst giving this another frustrated attempt to fix it. Tediously readded all the devices in my home with settings and automations. Invitation issues persisted.
Noticed that I didn’t have all my options for HomePod in Home app for it’s settings at this point. That was a new issue. As initially stated, the HomePod has been giving me issues since 11.4. Not giving me the sound effect when activated, spotty voice ”hey Siri” detection compared to it’s first few months, etc. This was a huge tell and coincided with the invitation issue appearing here.
I removed the HomePod accessory from my Home app and then hard reset it. This immediately allowed the other person to successfully be added on both their devices. Accepted the invite on iOS device and it synced in the background over iCloud and loaded on the other. No longer asking to accept invitation on the other iOs device owned by the guest with the spinning wheel of failure.
Another person in my family never had an issue on their iOS devices. I’d recommend looking at any home hubs and signing in and out of the them. It would be a pain to reset an Apple TV, but worth exploring. This took way too long to figure out and senior engineers couldn’t fix it. My hunch is that iCloud sync with regards to 2 factor isn’t pushing through on the backend on the hubs during a slightly corrupted upgrade on an Apple TV 4 or 5 and HomePod. No one has been having luck in my travels signing out of iOS iCloud accounts.
All is well here now.