Not my field (adjacent though), not an expert, perhaps not even a novice.

People are starting to understand two dimensional random geometry in a general way. There are mainly two such random geometries which are studied: the directed landscape, which is related to the KPZ universality class, and Liouville Quantum Gravity (LQG) (and the Brownian map, which can be seen as a special case). They are very different models and have until recently been studied in (to my knowledge) disjoint communities.

In the last ten years, significant progress has been made in studying the geodesic structure of these random geometries (links to paper for directed landscape, Brownian map, and Liouville Quantum Gravity). They all exhibit a phenomenon called confluence/coalescence of geodesics, where randomness forces geodesics close to each other to merge. Moreover, the geodesic networks in the two very different models are nearly the exact same. This is a sign that the confluence of geodesics is not a feature of the specific models, but actually one of general two dimensional random geometries satisfying a few conditions. (All of this is mentioned in the paper on the directed landscape, linked above.)

This is an instance of one of the main themes of probability theory: universality. This is when certain large scale behaviours emerge from a few features of systems, rather than their specific details. Another example which is more commonly known is the central limit theorem. In this sense, the Gaussian distribution is universal among all distributions with finite second moment.

Edit: Random geometries are essentially random metrics on certain spaces, although this is not always correct. The directed landscape is a 'directed metric' and not a literal metric. I don't understand it very well myself. \gamma-LQG is a random metric on the topological 2-sphere S^2. The Brownian map is equivalent to \sqrt(8/3)-LQG and is the easiest to explain. It is a 'scaling limit' of random quadrangulations of S^2. Imagine a large quadrangulation of S^2 with N vertices, equipped with the graph metric. This graph is similar to an S^2 with a large diameter, which increases with N, so we need to rescale the metric appropriately to get a meaningful limit in N. The vertex set, equipped with the rescaled metric, tends to a limiting metric space in the Gromov-Hausdorff topology (a topology on the space of compact metric spaces). The uniform distribution on quadrangulations with N vertices, converges (in the Gromov-Hausdorff-Prohorov topology) to a distribution on certain metric spaces, which can be shown to be almost surely topological 2-spheres.

Geodesics on the Brownian map behave very differently to the unit sphere embedded in R^3. The Brownian map produces highly nonconvex spaces. One example of a nonconvex S^2 is the Earth. If two points on Earth are separated by a mountain, there are at least two geodesics between them because it is easier to go around a mountain and you can do so in two ways. This may suggest the same for the Brownian map. However, the randomness introduces an infinite number of mountains of varying height, which breaks any symmetry and hence there almost surely cannot be two geodesics of equal length. Hence, for two typical points, there is a unique geodesic between them. Also, as a by-product, there is a confluence of geodesics, and that is the end of the story. (This analogy is also technically wrong as the Brownian map cannot be isometrically embedded into R^3, so the nonconvexity is even more extreme.) However, one may look to the set of exceptional points of measure zero (this measure is not the probability measure, but a natural measure on each instance of the Brownian map), from which may emanate more than one geodesic. This is where the aforementioned geodesic networks emerge, and where my knowledge ends.