Though they do share a whole lot. It would be like making a phone rectangular with a touch screen and a home button. Obviously everything inside your phone is completely original, but because they look vaguely similar, you've violated your twin's copyright.
It seems to me that the extent to which monozygotic twins do not have the same DNA would be comparable to the extent that clones would not.
Wikipedia indicates that the number of differences averages in the low hundreds, surely with such a low number there would remain plenty of parts that remain truly identical. For the purposes of a twin suing a twin for patent infringement, couldn't we just assume one of the many unchanged portions was patented, not the entirety?