Q: iCloud photo is of my spouse.

I'm not understanding.    Sure we're logged in.   I don't understand why when I go to System Preferences/iCloud and see my account with her photo.   I can change it, but when she does the same thing with her account, it will have my  changed photo.  

Hmmm, I just checked https://www.icloud.com/#settings.   It has my photo, but it doesn't have my icloud account, it has my iTunes account.

Different IDs.   Different cards set up in Contacts.

Interesting.   I went to my wife's Contacts - and it had the little head next to my face.   I went to her Systems Preference/Users & Groups/Contact Card, and it opened *my* contact.   I went to my computer's contacts and edited my contact.   I went to edit the profile picture, and see two photos that I can select.   One of those photos is labeled "fruux".   My contacts are on https://fruux.com/contacts/, so that we can share our contacts.   That costs extra money but may be the cause of our problem.   We need the same set of contacts.