Just about everything impacts the sound. In my experience, mass is the most meaningful, but I'm certain that the shape itself will contribute as well.
Just how much, and how to talk about it would require more careful observations and test instruments to be made for the control. Make everything the same except for the body shape, using wood from all the same lumber. Could make a nice collection
As far as wood overall, in Alembic basses and guitars, I find the neck is the most pronounced place the wood affects the tone. On a fretless bass, next is the fingerboard. On a fretted guitar or bass, next is the body wood (that's the core and the laminates).
I have been carefully studying the last several hundred Alembic basses and guitars, and I have learned to hear many things that were not obvious to me a few years ago. Coco Bolo is incredibly distinctive, as is Quilted Maple. Koa is unmistakable. A Mahogany neck gives itself away easily, but the body wood variables blur a little when paired with it.
In the end, the best advice is to select woods that won't fight against the default sound you are seeking to avoid a major custom blunder. For instance, an original all-Maple Essence is probably not a good choice for a typical country band, but can be stellar with rock. If you like the look of Koa, but the warm sound with less pronounced highs is exactly opposite the tone you are trying for, then you shouldn't use it except for a decorative veneer on the peghead.
Though I've watched hundreds of basses and guitars through the real time spectrum analyzer, and my sensitivity to changes in tone from wood has expanded, I'm always learning more every time I try. So please, listen to as many instruments as you can and if you're really interested, keep a notebook with your observations, it's very useful for reference.