You seem to be asking several different question.
"How can people be sure that the A4 note of a Piano is 440Hz?"
Based on the title I thought you were referring to person identifying the Hz of a tone by hearing it. In general, from what I've been taught, most people CANNOT identify absolute frequencies. We are very good at identifying relative frequencies, i.e. intervals. But it takes a special gift to Id the frequency by hearing. I read an article about 20 years ago that claimed people who speak languages that require tone for definition (specifically Chinese) are highly likely to have perfect pitch in adulthood even if they do not show musical talent or interest.
"I'm asking in a more practical/physical way. As far as I know, any notes on the keyboard are made of different frequencies, so A4 isn't made by 440Hz alone (if it does make up the sound)."
This statement hints at two things. The first being, is there a device that is capable of measuring frequency (or many frequencies). The second being that an instrument produces many frequencies to produce a tone. To the first point, yes there are devices that are capable of registering a frequency and they are very easy to make. Even before electronic tuners people could dissect the multiple harmonics in a complex tone using resonators. The idea is based on simple physics. For a complex system that produces waves we usually have many (possibly and infinite number) of harmonics all mixed up in the wave field. If you create a mechanical system that has one and only one resonant frequency, say a pendulum, or mass on a spring, and figure out a way to excite that with the acoustic field then it will only move with a large amplitude when its resonant frequency is present in the field. This type of experimental set up was used by Helmholtz to analyze acoustics. The method probably goes back thousands of years.
Now we have sophisticated electronic devices that can capture the full signal and dissect it using a mathematical signal processing algorithm called a Fast Fourier Transform.
"This is how you can have different instruments and say them all play A4 and yet they sound different, they're made of different frequencies with different combination of amplitudes (which makes up the timbre)."
This comment speaks to the fact that in complex systems a single excitation will generally produce hundreds or thousands of frequencies related to the vibrational mechanics of the system. And yes, the tone or timbre of the instrument, i.e. that characteristic sound, comes from two things (1) the mixture of harmonics that is allowed by the instrument and (2) the attack of the musician.
"Is there some sort of device or way that receives a sound wave and returns a single frequency?"
This may NOT be the right thing to do! It depends, the resonator I described will respond noticeably at one and only one frequency (Newton's second law says it will move in the presence of any force, but resonance means that even a small input at the right frequency produces a very large motion so there is really no chance of confusing the response). Since it is hard to produce a pure tone without harmonics this type of device is really the only thing that will output a single pure tone in response to an acoustic vibration. Modern electronic tuners probably respond to either the highest amplitude frequency in the complex waveform (which may NOT be the fundamental) or if there is some sophisticated circuitry in there it may be doing some signal processing to guess at the fundamental. Vibrating strings and resonant air columns in a tube come pretty close to obeying the "harmonic" relationship fn = n * f1. And we make instruments to follow this relation. If a {microphone + data acquisition card + computer} system picked up a wave and found 440Hz and 660Hz you would NOT have a harmonic relation, you would have a perfect 5th and that may mean that you truly have 2 fundamental tones. The only way to know for sure is some other data, like you are a piano tuner (the person not the device) and you know for a fact that you only hit one key while the others were dampened. Then perhaps these are 2 harmonics from the sequence (220Hz, 440Hz, 660Hz) and the real note being played was 220Hz. This can actually happen! There is some possibility for ambiguity in what is really being produced. The system could have been excited in such a way that the fundamental was very weak and the first two harmonics were strong, and the noise floor was high enough that the fundamental could not be determined in the spectrum. There is really no way to know. If you truly had two notes being played, 440 and 660 then you may have the harmonics of each intertwined. This produces a signal fingerprint and we can separate the sequences out, inferring that 440 and 660 were really each separately present. There are a couple things you should know about humans: (1) our ears are non-linear and produce aural harmonics when excited by a single pure tone. Thus is is not possible for a human to have ever experienced a single frequency, and (2) the brain has the ability to discover the harmonic sequence and determine what the fundamental is even if it's missing. This is called fundamental tracking. Because of this you can trick a human into hearing something that is not there, not present in the acoustic field and NOT created by any source. This makes human's unreliable in an uncontrolled environment.
Lastly, some vibrating systems do NOT follow the harmonic sequence, they have harmonics but they follow some other sequence. I have personally done research with the acoustics of such objects and can tell you that when you excite multiple harmonics you hear them as different tones rather than a complex wave. This is probably due to the fact that we have evolved to respond to the harmonic sequence and anything off that our brains handle differently.
When you FFT a complex wave form and try to infer the fundamental using some type of logic there are ways to get the wrong answer.
To answer your question there are in fact devices that will take a complex wave as the input and spit out a single result but they may either be too limited (like the mechanical resonator) or error prone.