That's an arbitrary definition. Milkshakes can be made with milk and ice cream and be thick as hell. A straw is a tool used for drinking, so I'd say if you can't drink it, then it's not a milkshake. But it doesn't hinge on a straw.
Edit: I think a blizzard could classify as a milkshake as you can drink it. You might have to give it a minute or two to defrost, but it is possible.