Nope. It's that there's only one lawful Functor instance. But Applicatives and Monads can be multiple - lists are the classic example (zip vs cross-product)
The cross-product is not to be confused with the Cartesian product, which is related to the (in this case unfortunately named) "cross join" in SQL. Cross products operate in ℝ³, while Cartesian products are just defined over sets. The standard List monad instance uses the latter.