Does the apparently dubious proof assert that there are exactly that many functions? Because if the assertion is rather that there at least that many, that assertion is true but really understating the case. E.g. for n=3 there are 2^(2^n)=256 functions, and after that it really blows up.
It gives that as an upper bound on the size of the search space of boolean functions with n-bit input, representing technical strategies that consider the last n market movements. It goes on to claim that if there is a historically winning technical strategy, a nondeterministic Turing machine can write out a full description of that strategy in polynomial time.
The remaining problem is how to uniquely describe in polynomial space an element of a set with doubly exponentially many elements, but the authors don't bring that up because they're working under the assumption that there are 2^n functions rather than 2^(2^n).
The assertion that the efficient market hypothesis is equivalent to P=NP also comes with no proof that future performance of a trading strategy will match its past performance or that anyone will actually deploy a perfect strategy if one exists.
The only "proof" of this I've seen asserts that there are 2^n boolean-valued functions on n-bit vectors.