Interesting question... I'm not sure to what extent you're aware of the literature, but have you looked into multinomial logit/multinomial probit models? As I explain below, my sense is that there probably isn't an approach that doesn't require some additional assumptions. However, just looking around quickly, maybe a paper like this one might be of interest? They propose a varying choice set logit model which seems quite closely related!
To expand a bit on my worry, I typically like to think of these choice problems from a utility framework, but put simply, in these choice cases, observing an individual choose good $j$ out of a vector of unordered goods $(1,\dots,J)$, all you know is that the individual prefers good $j$ to all other available goods. The available part is key: if all individuals choose over the same bundle, then the models I mentioned are quite popular, and there are some classification models that are also used.
However, in your problem, different individuals face different choices, and importantly, who faces which choices is not random (it's by country, which presumably differ). In such a case, when comparing two individuals from different countries, you cannot compare their choices because they are not choosing over the same bundle. Suppose country A chooses between goods (1,2,3), and country B between (1,2,4). You need some way to understand the desirability of good 4 to those in country A, but you never observe those individuals considering that choice. Using information from country B to inform country A choices about good 4 would require assumptions about individuals across countries being similar. And restricting both countries to those who choose either 1 or 2 will not help, because I'm then assuming something about how individuals choose that omitted good. For example, suppose that everyone in country B who choose 2 would have actually chosen 3 if they could have. Then by restricting to goods (1,2) across both countries, im comparing fundamentally different sets of individuals: those in country A truly do like good 2 over the other two options they faced, but those in coutry B prefer good 3, but only choose 2 because it was the second best option (the best that they faced).
This is quite rambly, but I have trouble believing that you can handle this structure without layering assumptions, and my guess is that they will all be quite strong, and I lack the knowledge of your problem to provide much guidance on that aspect... I'd suggest starting by looking into the models I mentioned at the start, and maybe see if there's any research about such models in the presence of restricted choice sets?
