One thing that's being ignored here is construction method. Is it a neck-through, set-neck or bolt-on. Those have a profound effect on the tone...
They do?
Most people are able to tell the difference in the real world, from my experience.
Having played a bolt on EBMM Ray back to back with a neck through EBMM Ray, Ive gotta say that the two basses sounded identical
The physics supports the differences heard, too.
This seems like a dubious claim without an oscilloscope and a controlled environment.
Everything makes a difference, to varying degrees.
Maybe, but with such small diminishing returns that it would be impossible to hear in a mix with other instruments
Throwing a humbucker in a Strat won't make it sound exactly like a Les Paul and vice versa.
Both my Strat and LP have Dimarzio Distortions in the bridge position and sound remarkably similar. Having swapped both pick-ups back and forth, I can tell via my DAW one pick-up is metering hotter than the other, so the majority of the variation in tone is likely between the two "identical" pick-ups. Also, the actual resistance between all the 500k pots that I use have a pretty substantial variance (seriously, if you've got a meter, check some of the resistances that get called "500k" - it's pretty wild what they actually read as) so I am sure that is causing all kinds of effects on the upper harmonics as well.
Set-neck is actually the one that involves the body wood the most. With both neck-throughs and bolt-ons the neck wood plays a greater part in the equation (although neck wood choice is still more important in a set neck than body wood). Bolt-ons tend to be more "snappy", on neck-throughs notes tend to exhibit more "bloom". Stay tuned and make some popcorn while I put my notes together for that flame-war starting thread one of these days. As a side-note, "sustain" is not significantly different between them, depending on how you define it, but the shape of the note decay is.
Even if the wood of the body makes an audible difference, it's impossible to attribute a single characteristic to a specific species of wood. The density in the grain varies SO MUCH that you could have a piece of alder be more porous than ash, while both being denser than a cut of mahogany.
Ash, by the way, is such a huge lie. Lumber mills will tag all kinds of wood as "ash" despite there being 100s of distinct species with distinct grain patterns. I was a purchaser for a mill, and I saw some seriously ridiculous claims from lumber suppliers.
At the end of the day, I'd say choose a wood species for a specific grain patter, color, weight and durability, but if you're looking for that "alder sound" you are likely going to spend a lot of time chasing windmills.