The most "postmodern" way that mathematics enters statistics/probability is probably via the category of Markov morphisms and chains like mentioned in
Cencov, Nikolai Nikolaevich. Statistical decision rules and optimal inference. No. 53. American Mathematical Soc., 2000.
As @Michael Hardy said, there is a functor between the category of Markov morphisms and the category of stochastic differentials. In fact, the simplest i.i.d. samples in statistics/probability is a Markov chain with constant transitional mapping. Also you can derive a corresponding SDE to describe an i.i.d. samples. In statistics, researchers tend to describe i.i.d. samples as outcomes of experiments. In this sense, statistics/probability is a variant of mathematics. The work of analysis of paths of a stochastic process mentioned by @user190080 is founded as early as 1990s by Stroock using some primitive geometric analysis techniques arxiv. But later his interest shifted.
A less elegant way is developed by Amari et.al, they introduced a geometric connection called Amari-$\alpha$ connection over the collection of statistical models(i.e. probability measures) to make it into a manifold in sense of classic Riemannian geometry. A related discussion is matheoverflow.
This approach seems very elegant at the first glance, but later on people figured out that the "informational geometry" is difficult to use since it is easy to use geometry to describe statistical models yet it is hard to transplant results in geometry to statistics/probability.
Mumford and Michor later applied differential geometry (mainly the study of the manifold of differentiable mappings) and Hamilton approach onto a branch of statistical shape analysis. Their work is fruitful yet still not yet a mainstream.
A quite interesting introduction of topology techniques (basically weak topology of Polish spaces) into statistics is the arise of Robust statistics around 1980-1990s when Huber and his students spent a whole lot of time at Princeton promoting their robust scheme(Princeton Robust Study). Now this branch is no longer heated but there is still some accumulative new works.
Today, in the study of "data analysis", persistent homology entered as a computationally convenient tool to capture some feature of the data, they are very popular in multiresolution statistical models. Wiki
There is still a long way to go before establishing a complete functorial correspondence between statistics and postmodern mathematics and I do not think any theoretical step is easy at the current moment given the overly heated trend of "statistical learning" in the community.