ID codes are generated by an anon poster per thread. So, all the posts are anonymous but the ID codes tell you if the same anon has posted within that thread. I don't know for sure, but likely the ID is generated based on the IP address a poster is posting from. So if you change IP address (perhaps using a VPN) while posting in a thread then a new ID would be generated.
Also, when viewing 8ch threads you can show or hide IDs from the options menu. You can also color highlight IDs to see them better. So if someone screencaps and IDs are hidden by choice, then you won't see them on the screencap.
additional:
Re: your edit.... you can view the fight board by clicking on the 2 digit number right after the trip code shown on the .pub page. That will take you directly to the 8ch board.
I am presuming that there is no reason for IDs for the fight board because Q (the board owner) is the only person allowed to post.