The manufacturing process results in a cylinder of silicon. The manufacturers could just square it off and return the trimmings to the pot, but as the whole process evolved to handle round wafers they just don't.
So the direct answer to "Is there some aspect of the lithography process that requires that the surface be round?" is "the machines are designed to accept round wafers, so that's what the wafer makers deliver. The next machines need to work with existing round wafers, so ......" and on it goes.
Some manufacturers will make smaller chips in the corners, it really depends on compatibility and economics. I have heard of foundries specializing in custom imaging elements (which can be quite large) adding smaller imaging devices for "free" around the perimeter. "Free" in this case being far less money than usual.