Well in my experience I think it does and it does make sense to me. I mean as a Gibson fan who's had the good fortune to play through (but not necessarily own) the wildly varying styles of Gibby bodies I notice a marked difference even though the hardware/electronics are more or less the same (tune-o-matic bridge, chrome 57 pickups, standard Gibby pots/wiring, etc). Like an SG is kind of light and breezy sounding with just enough crisp and just enough dark but a little thinner than other models. A Flying V is dark, full and bottomy (as is an Explorer from waht I remember but I do not have much experience with those). A Les Paul is surprisingly crisp for it's density but has a nice full midrange (I actually don't like Pauls but they certainly are unique and cannot be replaced if you are going for that specific tone).
Those are all solid bodies which with similar hardware and pickups should behave sonically exactly the same way if body shape had nothing to do with it. I have also noticed that these characteristics seem to hold true even on clones.
Other brands are harder to tell the difference on because... well there aren't that many wildly varying body styles with Fender and you likely won't get the same consistency amongst hardware/electronics with other brands.
That said... I'm no luthier, sonic genius or even a particularly intelligent individual. It could all be psychological but I notice a difference and actually seek out body styles based on these experiences because I have to rely on clones due to extremely limited finances. For example... I currently have a strat style Pacifica with a Hot Strat pickup config that does indeed behave like a Strat when I tell it to (but because of the humbucker can be summoned to do the devils work as far as nice crunchy metal rhythms/leads). I also have an Ibanez Les Paul clone that despite the absolute arse quality of it and the extremely underpowered (non-chrome) humbuckers can be coaxed into doing the job of a Les Paul.
What I would REALLY like is to have a good SG style again and an old ES style (Epiphone has a reasonably priced ES series on the go right now I'd love to get my hands on but the first electric I owned was a Saga II semi hollow body ES clone and it was brilliant in all sense of the word).
That said I also own an ancient Ibanez roadstar that is Strat style that no matter what I load into it will never sound like a Strat but I think that has more to do with the wood used in the neck and the body and the ridiculous amount of lacquer on the body (I swear it's like over a millimeter worth of coating). It is however thicker in the body than a real strat and is in general shaped differently as far as cutaways and subtle curves. It's hard to explain without picking it up but it's just bulkier and an example is the horns at the neck cutaway are just far less sculpted and much thicker likely deadening a lot of the tone and/or fattening it up. Still a very nice guitar and is more suited for metal with the right pickups.
tl;dr... yeah, I think shape has a lot to do with tone.