I love zebras

The description of the case doesn't make sense to me, either. But I'm not having an easy time imagining what the philology example would be like, which makes me worry that there might be something specific about the philology example that would affect things. I presume there's a reason you're avoiding giving too many details on purpose, but if you only reconstruct the features of the case that you consider relevant, and the case doesn't make sense to you, it may not be very revealing that the case doesn't make sense to others; the problem may be that something you are treating as irrelevant and not mentioning actually matters.

I mean, if I imagine that we're looking at words in a body of literature, and horses are word A and zebras are word B, and the argument is intended to show that word B was actually in use in that body of literature (as opposed to only appearing as a result of slips of the pen, copyist errors, etc.), then I can't see that this statistical argument proves anything; what we'd really want is some data about how common such errors are and what forms they usually take, in order to determine how many errors would be likely to arise by chance. Comparing to a hypothetical no-error case seems, as you say, a red herring, entirely and bizarrely beside the point. But this is also so obvious that I find it hard to imagine that anything like this was the original argument. Perhaps I am being too charitable.

The main reason I posted this is that I am sometimes wrong about things. Maybe the zebra example turns out to make sense in some way I hadn't thought of. Maybe Yagami is using some sort of standard method. Maybe there's some failure mode I haven't thought of. It would be really good to know this before I make an ass of myself with the review. And talking about asses - there are some wild asses in Mongolia which got left out of my parable - but they're kind of cute so here is a link.

The test is about whether or not the equids being real correlates with them being horses. It concludes that there is a correlation. An equid is more likely to be a horse if it's one of the ones you made up.

I'm thinking the writer was not using the appropriate test for what he wanted to show.

I think I agree with you that this is bogus statistics.

It sounds like the underlying claim is "there are wild zebras here and this is interesting." And the question isn't how many zebras were observed compared to the number of horses, the real question is the probability of recording a zebra when you just saw a horse with dust on it. Which yes, Fisher's exact test won't tell you.

My sense is that many researchers are confused about statistics, and so the prior for "author is confused and the statistical machinery is being misapplied" should be high.

I agree wholeheartedly with OP's conclusions.
Here, AFAICT, there are two uncorrelated problems.
The first is that Fisher's test can be used to show correlation in a multivariate distribution against the null hypothesis of uncorrelated distribution. Here clearly we do not have such a distribution, since the 'virtual' example is actually the prior, or the null hypothesis in frequentist parlance. But you cannot inject the prior in the data sampling distribution, pretending it to be a "virtual" parameter, and hope to get meaningful results back.
The second problem is of course that the prior is categorical about the absence of zebra, an assumption clearly refuted by the data.
Yagami's conclusion still stands, but for a much simpler and certain reasons than his bogus analysis: if you assume there are no zebras, and you find some zebras, then the assumption is refuted, with certainty 1 (or again, in frequentist-speak, with p=0).

I don't understand the point you're arguing against, so I can't evaluate your argument against it. What's the conclusion being drawn, in terms of what Dr. Yagami anticipates?

I agree with Apprentice: this is not a comparison of two independently obtained samples but an observed sample vs a predicted state of affairs.

The problem with this situation is another flavour of 0 and 1 are not probabilities. Here, treating the expected population of zebras as literally P(zebra)=0 in which case inferential statistical methods related to possible variation around observed values break down. Under the null hypothesis: observed data come from the expected distribution, and if P(zebra)=0) the variance in this distribution = 0.

Or put a different way, the "expected" distribution is not a distribution as we typically consider them - because 0 is not a probability.

It might make sense if there were several groups of people who independently counted the equines in Mongolia. 9 of the 10 groups did not encounter any Zebras, but the 10th did. What Dr. Yagami is suspecting is that the 10th group did not actually count equines in Mongolia but got off track and counted somewhere else. An alternative explanation would be that they did in fact count in Mongolia but since they were much more eager than the other groups they discovered the few Zebras in Mongolia which the other groups didn't. So what Dr. Yagami is trying to prove is that the 8 Zebras are too much to be a statistical fluke that the other groups just didn't get and it is more likely that the 10th group did in fact not count in Mongolia. Although I'm currently too drunk to figure out whether that is actually supported by the narrative.

Turn the question on it's head and make up a story where the math matches the observation in every circumstance. If you can, and I'm not sure I could, work backwards from there to find the breaking point. Or just remember that the map is not the territory, the finger that points to the moon is not the moon, and get on with things.

IMAO Dr.Yagami doesn't understand statistics and is spectacularly confused (did a zebra kick him in the head, by any chance?) about p-values and significance testing.