The percepted difference between, say, a C from a violin, a trumpet, and a singer, is due to the sounds having different overtones, that is notes at various integer numbers of octaves, thirds, fifths, etc. above the tonic (the "basic tone").
Sound is compressions in the air recieved by your ear (or a detector), and can be pictured by a plot showing the amplitude as a function of time.
If instead you plot the amplitude of the different frequencies, you have a so-called Fourier spectrum. A clean C from a synthesizer would just be a delta function (a single peak) at that note. But a C from, say, a violin would have several peaks at higher frequencies.
So the answer to your question is: yes, each human, animal, instrument, etc. have unique sounds, and there do indeed exist programs (e.g. the app "n-Track Tuner" for iPhone) that can Fourier transform a sound for you (of course, many sounds are so similar and/or fluctuate so much in time that distinguishing them can be difficult).
Below are two pictures from that app, the first showing the Fourier transform of tapping my coffee glass with my key (which turned out to be an A), and the second one whistling the same A.