I find myself wondering "why" for this particular aspect of Japanese, but that's probably a foolish question.

A maybe more relevant question - in English, you can say "I have two!" to indicate that you have two of something (well, it's usually clear two of what from context). In Japanese, can you say that, and if you do, does it have the counter word in it? So you'd be saying "I have two flat things" even if you aren't saying of what?