But in almost all cases that doesn't make any sense? Typically the data in different dimensions will have different "units". So there isn't any meaning in the scale in the first place. How could scaling by a single scalar be "more natural"?
It's true that there is no intrinsic meaning to the scale, but you must specify at least a relative scale -- how you want to compare (or weigh) different units -- before you can meaningfully cluster the data. Clustering can only work on dimensionless data.
In fact the paper doesn't assume that your dataset is contained in a vector space at all. All you have to give a clustering algorithm (as they define it) is a set and a metric function on it.
(the paper if you don't have a link: https://www.cs.cornell.edu/home/kleinber/nips15.pdf)