$\bullet$ First, heavy-tailed/Pareto distributions (aka power laws) frequently occur outside of finance and economy. They are used to describe the size/height/magnitude/severity distribution of countless natural phenomena.
A few examples are ocean waves, volcanic eruptions, asteroid impacts (a well-known example is the size of the craters on the moon), tornadoes, forest fires, streamflow and floods, solar flares, landslides, rainfall...
Other interesting human-related examples are the size of human settlements, size of files transferred on the web, Google's PageRank numbers of webpages...
This is closely related to the field of extreme value analysis (hundred-year floods, rogue waves...) - see the answer by Joel listing some famous extreme value distributions.
It is also to be noted that the empirical distributions (histograms, KDEs) of the phenomena above are heavy-tailed. In other words, we already observe the power-law naturally emerging in these phenomena. So yes, we do use parametric models to approximate the naturally occurring distributions, simulate values, etc., but I don't think we can say that it is artificial. (The paper Clauset et al. 2009 linked to by Sycorax in the comments above seems to be a good reference here.)
$\bullet$ Second, as far as your question, while power-law distributions are frequently observed, the underlying physical processes by which they arise are still mostly unknown. Indeed, natural phenomena are complex. E.g., floods are produced by the interplay between meteorological and hydrological processes, but are also influenced by infrastructure (e.g., dams) and human activities (e.g., land use). Nevertheless, there are many underlying processes that are believed to generate fat tails in the distributions of natural and other human-related phenomena, such as:
Newman, Mark EJ. "Power laws, Pareto distributions and Zipf's law." Contemporary physics 46.5 (2005): 323-351.
El Adlouni, S., B. Bobée, and T. B. M. J. Ouarda. "On the tails of extreme event distributions in hydrology." Journal of Hydrology 355.1-4 (2008): 16-33.
Mitzenmacher, Michael. "A brief history of generative models for power law and lognormal distributions." Internet mathematics 1.2 (2004): 226-251.
Clauset, Aaron, Cosma Rohilla Shalizi, and Mark EJ Newman. "Power-law distributions in empirical data." SIAM review 51.4 (2009): 661-703.