Could you please check you are using the latest release of the platform on this? I cannot see any issues with this when testing on my end.
Here is what I did, just to ensure we are on the same page
Edited one of my test forums and set it so that people could post anonymously
Created a test group which allowed anonymous posting
Created a test user, who was a member of that group
Posted a new reply to a topic as that member, and posted it anonymously
Edited my own group (now logged in as a member with full admin and moderator permission) to ensure I do not have permission in there to post anonymously in the group setting
Edited the post I created above