What makes you believe that the sound of an instrument is any less unique than a voice? You can play an A on a trumpet, a piano, a clarinet, a guitar, a voice, etc, and still be able to figure out which instrument it is. The fundamental frequency (note) is the same, but the specific pattern of over/undertones and harmonics, (timbre, tonality, etc) is unique to each type of instrument.
You used the phrase “mimic”. As other people have stated, a recording isn’t “mimicking” anything. Think of a photograph. The camera doesn’t guess what the image is and tries to reproduce it. It literally uses sensors to capture exactly what’s happening in front of it.
Synthesizers, on the other hand, do “mimic” instrument, because a true synthesizer creates the sound using electrical components and/or algorithms to create their sound, instead of using a recording.
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