That said, the broad body (that's what BB stands for)
The "broad" in Broad Bass intially meant that it was versatile, i.e. good at producing a broad range of sounds. Someone linked to a page from an early Japanese catalog (I think it may have been @hypercarrots?) in a different thread which lays this out.
But it definitely does have a broad body. At first glance it's obviously influenced by the Fender P bass body shape, but it's got a little bit of a fatter ass. Compared to my Fender, when I play my BB735 my hand tends to want to rest a little closer to the bridge - I think this is a result of the body shape.
I absolutely agree that they're great basses though!