what aspects of the sound can be measured and used to define it...
... prior to going in ones ear and becoming timbre.
What makes the sound identifiable is the perception/memory bit perhaps
Based on what I remember from college, the big factors going into timbre are the gross harmonic structure of the instrument's sound (i.e. the loudest harmonics / partials), the overall envelope (ADSR), how quickly various partials decay after onset, and the mix of harmonic / inharmonic content in the sound.
You can probably find better references that go into detail on this.
These things are all measurable but not easy to quantify in simple numbers.
For example we can measure distortion as a series of harmonics that all have individual amplitudes. We can see each harmonic, but it's hard to say what this sounds like just by looking at it. To make things easier we sum this up into one number as THD.
A measurement of timbre is a lot like a measurement of THD in that you will come up with a bunch of components, each with different amplitudes that change over time. However, unlike with distortion, there is no straightforward and useful way to sum timbre up into a number or even a few numbers.
There will often be 50-100+ audibly important partials in a sound, all of which you might find are impactful for its timbre. And each can change amplitude every few milliseconds. And all of those individual tones varying moment to moment ARE the timbre. And you can measure all that just fine, but it's hard to generalize about it.
Timbre is just a word for the overall sound of an instrument/voice, so the simplest way to experience it is to listen.
If you would like to play with timbre a bit, I can recommend downloading FL Studio (free demo) and loading some sounds into the Harmor plugin. It breaks audio down into those components and lets you manipulate them quite a bit. You can also watch a real-time visualization of each partial coming out of it. So this will tend to illustrate the concept for you in real time.