I'm not sure it affects how you might think about probability distributions in general, although it might. I think it's more helpful in thinking about inferences about distributions and models?

The page mentions one concept, for example, pertaining to uniform distributions. We usually think of an uninformative prior distribution in Bayesian inference as being uniform, which it can be, but what you probably want is a uniform prior with regard to a metric of parameter distinguishability or something like that. This will often change depend on the model, and might not be uniform with regard to the original metric of the raw parameter.

There's a lot of other uses an implications along these lines: how you compare and evaluate models and parameter estimates.