i don't agree (with steinberger) for the very reason that the unique sound of a berger is so very different from a wood, full-bodied instrument.
the wood of the body of an instrument is going to respond to the vibrational energy of the string. how it responds determines how much energy the string will retain at various frequencies as it vibrates, and how much vibrational energy will be transfered to the body, and then to the pickups connected to the body (bear with me here).
this can be experimentally shown.
i've brought this example up in another thread on this topic. if i play my doubleneck with the non-played neck -on- and the played neck -off- (i.e. i'm playing the fretted neck but the fretless neck is what's going to the amp) i can hear what i'm playing - very reverbed out, but still it can be heard. this is from the body vibration of the doubleneck - the huge body being vibrated by the strings i am fretting, moving the pickups with respect to the (relatively) stationary strings of the fretless neck, and causing what i'm playing to be heard.
it's very similar in sound to the end of zeppelin's "over the hills and far away", and i actually got the idea to do this from that song, or more specifically a guy i used to play in a cover band with who played that song on his doubleneck and would play the end with the 12 string neck on, but fretting on the 6 string neck.
this would not happen if the body was a non-issue in the sound of the instrument. that vibration, caused by the body vibrating the pickups in sympathetic frequencies to the notes i'm playing, moving with regard to the (relatively stationary) strings, which are affected very little by the vibrating body due to their own inertia and the very small vibrational energy that the body is exerting back on them. if the pickups move and the strings are stationary, as far as the pickups, and the amp, is concerned, it's the same thing as if the strings moved and the pickups were stationary.
note : this response is also not caused by the strings on the fretless neck vibrating in sympathy with the fretted neck because then they would only reproduce open string notes, not everything that i was playing.
here's an experiment that anyone can try that will illustrate this effect too, although not as well. get a cordless drill, vibrating hand massager, or any other battery-powered vibrating device you might have handy (

) and hold it against the body of your instrument (plugged into your amp) as you turn the device on. make sure it's far away from pickups, so as to not affect our test. hand-mute your strings over the pickups themselves, so they don't start vibrating. hear anything through your amp? that's the pickup vibrating with relation to the stationary strings you're handmuting.
use the same setup but have the device vibrate against other instruments. i bet you'll hear noticeably different sounds from the amp from the different instruments. might not be -hugely- different, but it'll probably be noticeable.
that "not-huge-but-noticeable" element is the different effect on the over all sound of the bass the body wood and construction of one instrument is having when compared to another.
a note about vibrating pickups - although pickups are generally not mounted directly to the body without some kind of acoustic coupling, the vibrating body will still cause the pickups to vibrate, as is evidenced by my doubleneck experiment, or the "vibrating body" experiment i gave above.
if the pickups are vibrating, and the string is vibrating, there's also going to be phase issues between the two, where the pickup is going to vibrate certain frequencies in phase with the string, increasing their volume, and certain frequencies out of phase with the string, decreasing those frequencies' volume in the resultant output. these effects are going to be directly related to the different materials of the instrument's construction, since those materials are determining not only how the strings, but also how the body itself, and therefore the pickups, respond to vibration.
these effects are going to be very small, but then again, nobody pickups two basses and expects either of them to sound -that- different from one another - it's not like one's gonna sound like a whale and the other's gonna sound like a clarinet.