If I start on a musical note on the piano and play out the folllowing harmonics/overtones I can find all the notes in the chromatic scale via the harmonic series eventually.
That's true in one sense of the word 'note', but not in the sense that's relevant to your reasoning. The harmonic series contains harmonics whose pitches correspond to all the note names in the chromatic scale. So in that sense, yes, it "contains all the notes". However, it does not contain all the notes in the sense of containing all the pitches in the chromatic scale (e.g. over a range of octaves).
Have a look at the table here. That's a list of all the notes in the chromatic scale (within some range, using some reference pitch, and some particular temperament).
Does the harmonic series based on any fundamental contain all those pitches? No, it doesn't.
Does that mean that every note on the piano contains all the other notes via its overtones?
No. The harmonic series will contain notes of all the different names. It doesn't contain "all the notes", in the sense of all the actual pitches that correspond to notes in the chromatic scale.
what makes a note unique?
Simplistically speaking, and assuming that you are interested only in pitch and not timbre, the thing that makes a note unique is its fundamental frequency. If you're talking about timbre, I refer you to Dave's answer :)
EDIT: OK, so... pitch perception, simplified version.
A sound is a variation in the pressure of the air.
That variation in pressure can be depicted as a waveform.
Not all sounds have a particular 'pitch'. A sound that has a pitch is one whose waveform repeats, like this:
We can see that the "one cycle" marked by the arrow repeats again and again.
Now, the pitch of the sound that we hear depends on how frequently the waveform repeats - we call this the frequency.
In the diagram above, the cycle is about 0.0033... seconds long. So the frequency is 1/0.0033 - Which is 300 Hz.
And that frequency of the sound is is, simply speaking, the answer to your question - that's the thing that governs the pitch that we hear.
You might be confused as to why I haven't mentioned harmonics at all. But actually, to understand the idea of frequency, you don't need to understand the idea of harmonics. You only need to know that a sound wave that repeats a certain number of times a second has a certain unique pitch.
So why do people talk about the harmonics of a note, if they have nothing to do with the pitch?
One reason is that the harmonics are important in how the ear recognises frequency. Our ear can't look at a graph, the way we just have.
What the ear does is make use of a different way of representing a sound - not as a pressure waveform changing over time ("time domain"), but as a set of different frequency components ("frequency domain").
The best way to understand this is to try it yourself. Go to https://meettechniek.info/additional/additive-synthesis.html and play with this:
The above shows a wave that has only the fundamental frequency - no harmonics (this shape is called a sine wave).
If we add more harmonics, the timbre of the wave changes, AND the shape of the wave changes:
In these pictures, the graph at the top is a time-domain view of the shape of the waveform, while the sliders at the bottom show the frequency domain view - how much energy is in each harmonic. An important thing to notice is that the harmonics are all multiples of the fundamental frequency. H1 is 440 Hz, so H2 is 880 Hz, H3 is 1320 Hz, and so on.
The ear can measure these different harmonics as there is a special structure called the Basilar membrane allowing it to do this:
So how does the ear know what the fundamental frequency of the note is from the harmonics?
What it does is look at the frequencies of all the harmonics, and then works out the highest common factor of the harmonics. So if it hears harmonics at 125, 250, 500, and 625Hz, it will guess that the frequency of the note is 125 Hz.
Because of this, notes with very different harmonic structures can still have the same pitch:
Those two arrangements of the harmonics sound very different, because the relative strengths of the harmonics change the timbre of the sound, but the pitch is still the same.
One things those diagrams don't show very well is the change in frequency (the X-axis in those diagrams is phase, not time).
This video shows much better how the pitch of the sound we hear depends on the frequency of the wave (how quickly it repeats):
So, putting this all together:
Let's say the ear hears these harmonics (all Hz):
440, 880, 1320, 1760, 2200
What frequency will the ear hear?
It will hear 440Hz - or A4.
What about 880, 1760, 2640, 3520, 4400?
It will hear 880 Hz - or A5. (Note that's still A, but a different 'A').
And what about 98, 196, 294, 392, 490?
The ear will hear 98 Hz, or G2.
So I don't think it's really useful to think that a note 'contains' other notes because of its harmonics - that's not how the ear works. It groups the harmonics into sets related by frequency that we then hear as single notes.