Quick update: I just enrolled a fresh machine into the JSS using a new enrollment package I created to test. That machine has the Self Service icons show up as expected. I also tested a few other machines that were previously enrolled and the icon issue is hit and miss. A few of them show the icons as expected but others have the problem where the icons won't show up. Is there a client side cache that needs to be cleared perhaps on these problem computers? I already rebooted one to see if that would help but it did not.

Yes, you'll likely need to delete the local cache for Self Service on these systems. See this post for the location of the cache. Delete while SS is closed and relaunch to see if the icons comes back.
https://www.jamf.com/jamf-nation/discussions/11976/self-service-login-page#responseChild69154

I went to /Users/[username]/Library/Caches/ and deleted the whole folder for "com.jamfsoftware.selfservice.mac"

I then relaunched the self-service app and that folder was recreated, but the icons still aren't showing as expected. Is there also a global cache that isn't user specific maybe? I logged in as another user on the same mac and the icons don't show under a different user either.

@jerdill Did you ever resolve this? seeing the same thing after jamf 10 upgrade

Just built a Dev Jamf Pro 10.2.1 box and I'm seeing this same behavior. Deleting the cache files/folder listed above didn't help.

We are seeing the same issue after upgrading to Jamf 10. Not only are the icons and display missing from Self Service .. but they are also missing from the policies if JSS (Under self service tab). I can up load new icons and set Display name - but all existing Self Service info is lost.