I suppose the idea that no two snowflakes are alike could be tested statistically, but first you'd have to decide how different two snowflakes would have to be to be called "different." If two snowflakes were exactly alike except for one water molecule being in a slightly different position (which you couldn't see even with an electron microscope) would that be different enough?
If the average number of molecules in a snowflake is n, then the number of different configurations a snowflake could have would probably be on the order of 2n, which is probably an enormous number. It wouldn't surprise me if this number is greater than the number of snowflakes that could exist.
But I don't know whether anyone has actually done the math, or if they just made it up. I'd bet on making it up.