I don't know if this work is particularly ground breaking, but this looks to be more concerned with applications towards number theory (specifically cohen-lenstra heuristics) rather than adding to the category theory literature.
This is a bit like the recent news about the “ML discovered a new fast matrix multiplication” press push out of Google, where researchers in the field just sort of rolled their eyes.
Are you suggesting that this need to solve the moment problem all over again could have been avoided by searching Category Theory developments?
The use of moments as a method for understanding probability distributions isn't particularly new. What is new, is the introduction and adaptation of this framework for trying to understand the Cohen-Lenstra heuristics -- a smaller-scope problem in number theory that has seen a lot of attention recently.
The people quoted are among the world's leading experts in this topic, and I would say that Melanie Wood is the foremost expert. This is definitely novel.
The two authors are no slouches... I certainly wouldn't roll my eyes at them.
(I'm a specialist in this area of number theory)
This is correct. The word "category" is being used in its colloquial English sense; this is not work on category theory.
[Edit: I'm mistaken in the above point; see comment below]