All true. Any material developed and published by a legitimate licensee of CBS is owned by them. This includes books, comics, role-playing games, video games, toys, etc.
Besides, the "Battle of Axanar" was actually mentioned in the episode, so the name itself is owned by them, particularly in context as a battle fought within the fictional Star Trek universe fought between the Federation and Klingons. They might even have a case if another franchise tried to use the term "The Battle of Axanar."
Separately, what I find particularly interesting is this note from Memory-Alpha, regarding the fact that the TOS episode never specified who the opponent in the war was:
"According to the FASA source book The Four Years War, the battle was fought between the Federation and the Klingon Empire. This is also highly unlikely, as the history implied by TOS: "Errand of Mercy" indicates that the Federation had been involved in a strictly "cold" war with the Klingons during that time. In addition, Garth's reaction to the fact that Kirk seems to have successfully made peace with the Federation's opponents in the battle would exclude the Klingons as a potential enemy."
Based on the information, it could very well have been a race we've never seen, or someone like the Andorians, Tellarites or if you want to be creative, the Suliban.