That isn't how we process audio information, so no it is literally impossible. I can go into the acoustics if you want, but because of the way our ears/heads work, and the way our brain processes that information, we can't do that.
The simplest way to explain it, at least for me is to think about polyrhythms. When two different rhythmic figures are played over one another we never hear it as two distinct rhythmic figures, instead we hear it as the composite polyrythmic phrase. Sure we can focus on one of the two figures, but we will never hear that figure purely since it will always be a part of the composite rhythm. This basic idea also applies to pitch (and is the basis of western Tonal Harmony). Pitch is simply the regular rate of the compression, and rarefaction, of air. While the rhythm of say A440 is something we don't recognize as distinct rhythmic moments, that's exactly what is happening when we pluck a string tuned to that pitch. Likewise harmony is just our brain processing these rhythmic relations, which is why harmony with more extensions/notes sounds more dissonant to us, the compound rhythm is more complex. This is why when we listen to two songs at once it is a disorienting experience since it is likely that the two songs are either faster/slower than one another, or in a different key, or have some other structural difference. Because of this we will only ever be able to hear the compound relationship between the pieces, and never be able to hear them independently. This is much the same reason why music that is composed polytonaly is not really perceived as being in two separate tonal centers/keys and instead in a new unique tonal center.
This is most likely the case because the ability to individuate sound would have been an evolutionary hindrance in the past. It would have likely been more valuable for humans to hear things within the context of their surroundings.