> wine/water = (1-x)/x

It's symmetrical but not "naturally" uniform. If we consider both quantities as independent and uniformly distributed over [0..1[, then 1:1 ratio becomes much more likely to occur than 1:1000000. I find that approach "the most natural". The probability P(water/wine < 2) is 7/8 in that case.

The main lesson is that the Principal of Indifference is "not even wrong". (It's nonsense.)

So we are really looking at the density function of the hidden variable X?

For the original problem it feels like the interval constraints on the ratio are causing the issue since they will be inverse. But maybe I am not clear on that...

The "Grandfather" in the story argued that we can consider the problem specified if we assume that "the set of all distinct possible states" are uniformly distributed -- which is reasonable and correct when solving problems in quantum mechanics but unreasonable in ordinary everyday situations.

Thinking along those lines, you can construct a set of all possible concentrations: (0 wine particles, N water particles), (1 wine particle, N-1 water), (2, N-2), ... (N, 0). That led to my x/(1-x) formulation, which has the advantage of avoiding the paradox as stated, but I would not argue it's a well-founded result.

But more fundamentally, it seems a stretch to apply these quantum mechanical assumptions. One could interpret the question as "in all possible real-world scenarios in which such a test might be performed, how often would the concentration be in this range?" ... in which case one could consider all realistic models for how the mixture might have been created, and the likelihood of each of those actually being the case, and consider more and more things coming up with a more complete answer ... but one would never have a correct answer without actually stating some additional assumptions.