> Why do we call them product types anyway?

Answer in the piece is not wrong, but put more intuitively and succinctly: the total number of possible value (inhabitants) in a product type is the product of the quantity of inhabitants of the constituent types.

Replace the “product”s with “sum”s and it works too.

Interestingly, the total number of unique functions (judged only in terms of input and output) from a -> b can be found by exponentiation, (inhabitants of b) ^ (inhabitants of a).