You have provided a real example, which I was looking for, of why one might need to express a square dollar; thanks.
I wonder if the people who want to argue "types save you from bugs" see your example as very unwelcome, since they'd want to use "squared dollars" as an example of something nonsensical that should be flagged as a type error. I hope those people can reflect rationally on the limits of type systems in the real world.
A type system is merely a tool to encode information to help better model things. If you want to prevent multiplying dollars together, types can help you do that. If you want to enable multiplying dollars together, types can help you do that, too.
# oops my scaler has a unit
x unit * x unit = x unit^2
I don't know what that author would think of multiplying $100 with $20, but my point is that this embrace of type systems is apparently not just about function interfaces; it also includes the operands of things like multiplication, and preventing that operation if the types are fishy.
Using the example above, "multiplying $100 with £20" can be achieved just by making the better engineer a remote employee paid in pounds. It adds the exchange rate into the mix, so the math will change over time, but conceptually the math is the same as the "dollar * dollar" example.