Unfortunately, no. There are two impediments to such a program. First, there isn't a sufficient database of individual sound recordings. You might cobble together collections of certain common sounds from various resources, for example words containing /a/ or /ɑ/ or /æ/, but it is usually very hard to find recordings of a given putative sound with multiple speakers of a language. Finding multiple speakers for a language is important, because you have to be able to distinguish between peculiarities of a single speaker vs. peculiarities of a language. Taking three speakers to be the minimum sample necessary for making a somewhat credible claim about a single language, note that we still do not know truly language-dependent (as opposed to speaker-dependent) facts of VOT for the languages reported in Lisker & Abramson's classic work on VOT. Now, looking at Cho & Ladefoged 1999 which looks at VOT in a number of languages (all endangered, in a nice twist), they got data from a reasonable number of speakers of each language, and you can see that VOT for the aspirated / unaspirated contrast differs quite substantially across the languages that they sampled.
The second problem is even more profound, in that the underlying premise is false. That is, were one to listen to samples of the [k] vs. [kʰ] contrast across languages, one could never in principle get the correct answer by listening to any individual sound. Navaho unaspirated [k] is about comparable to Thai aspirated [kʰ] (both languages have a contrast between aspirated and unaspirated). This is especially clear in dealing with vowel height, where the "ɪ" of one language is the "e" of the next language. There are general phonetic ballparks, where a glottalized stop will have a certain range of known variation (and surely some additional unknown variation), but the idea that there is a single phonetic representation of a particular IPA symbol is clearly false. So to ask a listener to make an absolute identification of a particular sound is a conceptual mistake. What could be useful would be a comparative judgment of similarity between two sounds.
Substantial though still insufficient raw materials are available via the UCLA phonetics archive. A second possible resource is the Journal of the IPA, via their sketches, which nowadays tend to include online speech samples. Eventually, the materials will be available, and then what remains to be done is better defining what needs to be done with these samples, in order to better grasp the similarities and differences of phonetic segments across languages.
However... in light of the comment about expert performances, you can in fact get performances by Peter Ladefoged here, performing IPA symbols. These are not sounds of a specific language, rather they are reference sounds which may or may not match some actual language. You can also get productions of cardinal vowels by Jones here, and side-by-side recordings of Ladefoged, House and Wells performing vowels. The latter is informative in that it reveals variation within the community of well-known experts.