To compare - Running a M1 Mac Mini upgraded from Big Sur to Monterey and at very first boot as Brand New - it chose a Icon for User Account other than the Owl I use now.
At month later, purchased a M1 MBA also Big Sur as New( since upgraded to Monterey ) and it too was another icon for User Account. Not the same as Mac Mini but different.
Both Computer using the same Apple ID etc.
Neither computer synced the User Account icon to the other.
The only way I was to get them with same icon for User Account was by purposefully changing the icons to the Owl.
Same thing had to be done for my icon used on the ASC Forums.
So, I really do not really understand the iCloud icon issue you are referring to.
When I look in System Preference >> Internet Account and choose iCloud in there - I do not see any place for an icon / picture to be added or even a place to put an picture or icon.
Can one provide a screen shot of just where this is appearing on this computer.
If possible - make sure there is no Personal Information appearing in this picture. It would be removed by the Moderators of the ASC Forums.