Even if you don't ask for the ring structure to be preserved, this seems quite unlikely, at least if you require the functor to do the same "scaling up" operation to maps on cohomology induced by maps of spaces. I believe all the Goodwillie derivatives of such a functor would be zero, whereas "geometric" operations typically have some nontrivial radius of analyticity.

Even if you don't ask for the ring structure to be preserved, this seems quite unlikely, at least if you require the functor to do the same "scaling up" operation to maps on cohomology induced by maps of spaces. I believe all the Goodwillie derivatives of such a functor would be zero, whereas "geometric" operations typically have some nontrivial radius of analyticity.

Edit: For the argument I was thinking of, I need to assume that π₁(F(any sufficiently highly connected space)) = 0, so that for instance F(•) = •.