For example, could the iPad or iPhone accurately detect the difference between conductive pieces of plastic in the following shapes?
While it's still unlikely that such shape detection is possible (that would be so cool for boardgames or D&D maps or whatever), Ten One Design has successfully used private APIs to determine the size of the touch (which they refer to as pressure detection). http://youtu.be/OgTcyjzXfTg?t=32s shows it in action.
The touchscreen likely isn't sufficiently sensitive; there's a lot of very clever math going on behind the scenes to make it seems as smooth as it is. The actual resolution is something like 1/16 of an inch.
But worse, as @Dan points out, it's not really detecting shapes that way, the signal is much more crude.
A multitouch gesture recognizer can see when it's been touched in more than one discrete spot, and can understand the position of each of those touches. So that's "shape-ish". But no, there's no way to know a square versus an octagon is touching it.
The digitiser will track the number and location of touches, so if the shapes were distinguished by, say, conductive blobs on each apex then it should be possible to identify them individually.
There is no public API in stock iOS for detecting the size of a touch.