Why don’t they hardly ever call anybody by their real name? What’s the deal with the Saiho/ Taiho thing? Why didn’t they ever finish the Keiki trapped in the real world story?
I’m not sure why this OP has sat unanswered for nearly half a year, but here goes my attempt at a reply, having watched the anime series a while ago, and having just read the first novel in the series (where the plot is somewhat different).
There are two things going on here: some characters get names that go with their official positions, and those that get transported from Japan to the twelve kingdoms get a name that fits into the language of the twelve kingdoms. So, with the central character in the first story arc, Youko Nakajima, people won’t call her “Youko” or “Nakajima-san” as they would in Japan, because she is king of Kei (and should be addressed as such), and because she is given a name in the twelve-kingdom language, Sekishi.
The other name for these beings is “kirin”. It’s part of the whole government structure in the twelve kingdoms, which you just have to accept. The kirins are semi-divine creatures that have two roles: to choose the king of their kingdom, and to assist and advise the king in ruling the kingdom. They also suffer sickness and die if the king rules badly – but the king will also die shortly after that. I think it makes about much sense as any system of absolute monarchy.
Presumably both because it would be hard to kill a kirin – it would be hard enough to put an enchantment on him to stop him speaking – and also to allow his visible presence to give apparent authenticity to the false king of Kei. In addition, if they had killed him, presumably Youko (as the real king of Kei) might have died, and a new kirin of Kei would been born, outside the control of the false king of Kei.
(Note that I’m using the term “king” here, even though in many cases – e.g., both the false and the real kings of Kei – the ruler is female. The original Japanese term “Ou” does not denote gender.)
Sorry. I meant Taiki. At the end of episode 21 he’s trapped on Earth believing he’s a human being.
I should have checked Wikipedia. According to them the story ended due to the end of the anime.
Saiho (宰輔): The official rank of a kirin, that of second-in-command next to the king. See also ‘Taiho.’
Taiho (台輔): The formal title of a kirin. The kirin’s rank is actually Saiho, but out of respect, a kirin is referred to by their title of Taiho.
Why? 
Yes, they didn’t really finish that story arc properly. However, the whole thing is written to be part of a much larger history, so (as in life) you don’t get real endings. For example, with Youko Nakajima I really would like to get more of the story about how the kingdom of Kei does when she settles in on the throne, after dealing with the false king of Kei and the opposition among her ministers and governors.
I don’t know enough Chinese or Japanese to understand what’s going on there – the language of the twelve kingdoms obviously is related in some way through the written language, even if knowing spoken Japanese or Mandarin doesn’t help you understand it – but it’s probably pretty arbitrary, and just reflects the multiple levels of formality in Japanese, where things can have different names depending on the formality with which you are speaking or writing.