Rye is a kind of whiskey, sure, but whiskey without qualifiers almost always refers to whiskey made with wheat. In America you may get corn whiskey (which is distinct from bourbon in that bourbon has to be more than half corn in the mash) but I'll guarantee you could walk into a bar today and order "a shot of whiskey and a shot of rye" and the bartender would know what you meant and get you one shot of wheat whiskey and one shot of rye whiskey.

Also, the original meaning of whiskey is just "distilled liquor". It comes from a Gaelic phrase that means "water of life" and only came to mean what we understand as whiskey because that's how the people who created the term "whiskey" made their hard liquor.

It's an awful lot like saying "water and sparkling water". Sure, sparkling and flat are types of water but no one says "flat water"