it boils down to semantics...
two half dollars is still two half dollars, not one dollar. however, it equals one dollar. by the same token (no pun), they could each individually be a dollar, using the reference to US currency (coins and all), commonly referred to as "the dollar".
two half cups is still two half cups, and depending on which use of the word 'cup' depends on what those two amount to...more than generally, the assumption would be a measured cup...in which case, yes, they would amount to a cup. however, it could mean literally two halves of a cup ("container") that was cut down the middle.
back to the sandwich thing...as far as i go, half a sandwich is half a sandwich. if i ordered soup and a sandwich and i got half a sandwich, i would wonder where the other half of my sandwich went...i would think someone was crazy if they looked at a sandwich cut in half and told me there were two sandwiches there...even if there were two different bread type halves, i would still say there were two sandwich halfs, as opposed to "two halves of a sandwich" since that sounds like it implies that they were once one complete sandwich, together. or would you still say that's 1 sandwich, even if the two halfs are different bread, but same dimensions...
and then i think about this...what is half? is it still half a sandwich if you cut it 25/75? would you generalize and say you have two halves of a sandwich, and if so, then could you say you cut the sandwich in half, too? would you reference 'half' as 50% or 2 parts?
i don't think english is imprecise; in fact, it can be as precise as you make it (like code, it can be sloppy or exact, depending on you). what we suffer from is conveying imprecise (if that's the word) descriptions, and the meaning is interpreted based on your individual references and biases...
hmmm...
- g