What are those characteristics by which every sound can identified uniquely?
For example, pitch is one of the characteristics of sound, but let’s say a note C# can also be played on a guitar and piano with same pitch but the resulting sound that we hear is different so what are those characteristics which defines every sound.
For a long time, timbre was believed to be based on the relative amplitudes of the harmonics. This is a hypothesis originally put forward by Helmholtz in the 19th century based on experiments using extremely primitive lab equipment. E.g., he used Helmholtz resonators to "hear out" the harmonics of various sounds. In reality, the relative amplitudes of the harmonics is only one of several factors that contribute to timbre, and it's far from sufficient on its own, as you can tell when you listen to a cheap synthesizer. Flicking the switch from "flute" to "violin" doesn't actually make the synthesizer sound like a flute or a violin enough that you could tell what it was intended to be.
A lot of different factors contribute to timbre:
relative amplitudes of the harmonics
the manner in which the harmonics start up during the attack of the note, with some coming up sooner than others (important for trumpet tones)
slight deviations from mathematical perfection in the pattern $f$, $2f$, $3f$, ... of the harmonics (important for piano tones)
the sustain and decay of the note (guitar versus violin)
vibrato
Some sounds, such as gongs and most percussion, aren't periodic waveforms, in which case you don't even get harmonics that are near-integer multiples of a fundamental.
Because there are so many different factors that combine to determine timbre, it's remarkably difficult to synthesize realistic timbres from scratch. Modern digital instruments meant to sound like acoustic instruments often use brute-force recording and playback. For example, digital pianos these days just play back tones recorded digitally from an acoustic piano.
The entirety of the wave train is involved in the perception of timbre and it is not reducable to a few easily measurable features. There are entire books, including one by Helmholtz, investigating the characteristics of sound and how it relates to perception. I like Music, Physics and Engineering by H. Olson.
Often the power spectrum of a given sound is used to help analyze the timbre of notes; the relative magnitudes of the different harmonics, and the degree to which their frequencies differ from the theoretical ideal are a good starting point. However, these Fourier transform based approaches are not complete in that they typically do not address the perceptually salient features such as note attack and decay.
Another approach to take is to examine how synthesizers generate sounds; for this I recommend checking out the Sound on Sound article series Synth Secrets.
What makes a violin sound like a violin and not a trumpet, even though they are both sounding a note in the same pitch?
The answer is harmonic content.
A pure tone of a single frequency is essentially a sinusoidal vibration... repeated displacement back and forth which, if plotted against time, will appear as a "sine wave". When you drop a stone into water, and waves ripple out, the pattern of wave crest and trough is a reasonable approximation.
When a violin sounds a note, it is not producing a pure (sinusoidal) tone. Neither is any other musical instrument. There is a "fundamental", which is a pure sine wave at the frequency corresponding to the "pitch" of the note, accompanied by a set of "overtones" of other frequencies. These overtones occur at frequencies which are specific whole number multiples of the fundamental. The presence of an overtone at a given frequency, and its relative amplitude creates the harmonic structure of a given sound, and creates the sound you recognize as a violin vs a trumpet, or whatever.
Very few sounds you hear are regular vibrations at a single frequency (i.e. monochromatic sounds). Rather, most sound-producing objects will cause vibrations in the air at many different frequencies.
Mathematically, one can apply the Fourier transform to the function that describes the vibration over time. The result is a function that describes how strong each frequency component is.
Many instruments are decently approximated as consisting of a fundamental pitch and a discrete sequence of overtones on top of that, where the frequencies of the overtones are nice-looking rational (often integral) multiples of the frequency of the fundamental. Which overtones are present and what their relative strengths are comprise the primary components of the timbre (characteristic) of the sound. A C# will have the same fundamental frequency of $466\ \mathrm{Hz}$ or so on both a guitar and a piano, but the strengths of the overtones (such as at $932\ \mathrm{Hz}$ or $1398\ \mathrm{Hz}$) will be different.
By the way, if you have a broad continuum of frequencies present at approximately equal strengths, the result is known as noise (a technical term, yes). There are certain standard types or colors of noise depending on the specifics of the spectrum of frequencies.
Two waves never mix up together. They just add up when they meet while travelling along the same medium. This phenomenon is called interference. But they still retain their own properties and shapes, and return to their individual shapes when they divide. May be, brain uses this property to recognize individual sounds.
Look at these animations for better understanding.
