Is it theoretically possible to perfectly quantize a continuous signal?
Depends on assumptions you are willing to make and what type of signals are you trying to sample, but in theory I think that sampling rate equal to the Planck time would be a gold standard for anything...

This translates to sampling frequency of $1.855 \times 10 ^ {43} \mathtt{Hz}$ ($18.55$ tredecillion hertz). Personally I believe that machines will never be so fast. Obviously we can limit our assumptions to more realistic ones.

Obviously there is more constraints, such as: what sensor is capable of measuring displacement/velocity with such accuracy? Obviously this assumption is applicable even for subatomic scale, and most likely we don't need such accuracy in measurements of sound signals. Additionally when frequency is increasing, sound attenuation in media (air in this case) is increasing to enormous amounts and ultra-sonic sound waves are not propagating on large distances. At some point you will start measuring the Brownian motion. You also mentioned about possibility of measuring each particle separately and it is another, very difficult problem to tackle.