Good conversation, I think I'll jump in.
I would vote for the #1 thing that makes any electric instrument sound as is does is YOU. Your hands, the way you strike the string, even the way you hold the string to the fret. I've seen so many examples of this in my years of playing. That particular guitar player will sound like themselves on whatever instrument they play. "Look, Bob got a great new guitar ... and he sounds just like the old Bob."
The details WILL make a difference to you as the one playing the instrument. But just be aware they are subtle and likely undetectable by others.
This doesn't mean you shouldn't pursue the details. Partaking in "the quest" is not a bad thing, although it can be expensive if you fall completely down that rabbit hole. (Wait, who am I talking to in here? HA!)
One could certainly argue that the specific wood used on a Stradivarius makes a difference. The differences may be more profound for acoustic instruments whose bodies are built to project the sound. Although again, I would happily listen to Yoyo Ma play on a plexiglass cello if he decided to try one - or if one existed.
I know that Mica and the Alembic team have spent a great deal of time and energy working out the effect different woods have on tone production. I'm not arguing with that. Just suggesting - I believe this variable puts the "b" in subtle.
My own experience is this: I fell in love with the sound of a Series I 4-string bass in 1975. Got my first 5-string made in '76 and that was that! I loved the sounds I could make with these basses and never turned back. When I lost that original 5-string in '87 I had a replacement made with the same wood choices and the same dimensions. To avoid living through that loss again, a couple years later I had two more instruments made in the same style, one fretted and one fretless. The fretless was later changed to a fretted becoming a 3rd copy of my original.
So now I own 3 Series-II 5-string basses made of the same woods and with the same dimensions as my original 1976 model. But guess what? I have clear preferences between these basses. The one I like best is my recording bass. #2 is my road bass. And #3 is my daily practice bass which I also take on the road as a spare. These three "identical" axes are quite different, but I suspect only to ME. I don't think anybody hearing me play one or the other could spot this difference.
Now Michael D, I also have a graphite necked fretless which is pretty clearly a different animal. A through-body graphite neck, like aluminum I suppose, compresses the sound quite a bit. In fact, it compresses the dynamics to the point that I find it difficult to play soft and loud without moving the master volume control, if you see what I mean. It also sustains longer so the notes don't taper off as they do on my other basses - which I find ideal for a fretless. So that is pretty clearly a different sound. And yet I still get asked occasionally by people who should know better: "did you play fretless on that tune"? So while these basses are hugely different to me, it seems that I still just sound like me. HA!
I enjoyed that YouTube vid and appreciate the effort and almost scientific approach he took to the research. Good sustain on that air guitar too. Perhaps due to the 400lb bridge block?
Jimmy J