I have seen a computer experiment at a science museum that asked the user to distinguish very similar vowels by sound explaining that visitors with different native language can distinguish different vowels.
Is it possible in principle to identify someone's native language from their capability to distinguish sounds?
Different languages produce their vowels at different positions in the mouth. For example while English distinguishes two different a sounds (those in "lard" and those in "gaunt"), German has only one, which is actually positioned between the other two. However, unless you are an advanced speaker of a different language, your ears will not expect to hear this non-English a and instead try to file it as one of the a's that you know.
That is also why people have accents. Their tongues and ears are not used to the other sounds and often they might not even be aware of a difference between the foreign sound and a similar one in their own language. Trouble is, even though it might be similar, it's not the same, and native speakers of the other language can usually tell (because the sound is not exactly what their ears expected).
I'd say, to a degree it could thus be possible to determine someone's origin from what sounds they are able to discern. It would not be failsafe however, as you also have to take into consideration what accents and foreign languages the person knows to which degree. Finding things that the person cannot distinguish would thus give better feedback as finding things the person can distinguish.
Assuming the test subjects are not trying to fool the test (by intentionally picking the wrong answer or feigning knowing when they don't), then it's entirely reasonable that you can derive a reasonably confident list of distinctions the test subject is able to hear.
From that, you may be able to match that with trained (or manually curated) models on languages' expected matrices.
On the positive side, a test subject may be able to distinguish between lake and rake, but what would this tell us? Very little. But while it wouldn't allow us to confidently eliminate languages that lack that sort of a distinction, it might provide a strong clue.
On the negative side, a test subject may not be able to distinguish between lake and rake. This is slightly more informative as we might be able to eliminate many languages with distinctive contrasts of that sort.
Overall, I am strongly confident that you would be able to create a test that can be used to differentiate speakers of two drastically different languages. Given good statistical modeling, I am also confident that you'd be able to make a fairly strong n-way distinction.
This is an interesting premise, but one I think is fundamentally flawed. As a counterexample, in English, the main difference in pronunciation between regional dialects is in the vowels. Soundex was originally developed to get around this issue by deleting vowels in order to find names in lists (such as telephone books and census lists). There are some parts of the US where pen and pin are pronounced the same.
As an example in support of your hypothesis, Scandinavian languages have more vowels, and some of my relatives try to explain the difference between them to me, but I can't hear the difference. Is this training? Lack of musical ability on my part? Are they making fun of me?
This would likely be extremely difficult without considerable knowledge of perceptual patterns of a large number of languages, including social variation within them. At the moment, there is not adequate evidence for constructing such a system. In the U.S., for example, where there is substantial variation in the sound system, we've seen far less research on perception than production, and we know that the two are not always parallel.
