What is the correct notion of morphism between statistical manifolds? Given two statistical manifolds, is there a notion of "isomorphic"? What are morphisms?
 A: Any smooth (resp. $C^1$) statistical manifold can be embedded into the space of probability measures on a finite set.
A probability measure can be viewed as  a weakly averaging affine measurable functional taking values in the unit interval which preserves limits, so statistical manifolds can be viewed categorically as the same thing after their embedding into probability measures on finite sets.
From the abstract of the second linked paper:

The probability measures on a space are the elements of a submonad of a double dualization monad on the category of measurable spaces into the unit interval, and this monad is naturally isomorphic to the Giry monad. We show this submonad is the codensity monad of a functor from the category of convex spaces to the category of measurable spaces.

You could thusly take a morphism between statistical manifolds to be a morphism of monads between the appropriate associated monads in the bicategory of monads which commutes with the embeddings, and define an iso in the obvious manner.
Link 1: Lê, H.V. Statistical manifolds are statistical models. J. geom. 84, 83–93 (2006).
Link 2: Sturtz, Kirk, Categorical Probability Theory, arXiv:1406.6030 [math.CT] (2015)
