I'm only a fractal enthusiast, but my impression is that the key distinction that makes these fractals, or at least fractal-like, is not their detail per se, but that there is complexity at every scale. From the article:
> we find intricate structure at every scale
> At every length scale, small changes in the hyperparameters can lead to large changes in training dynamics
> we find intricate structure at every scale
> At every length scale, small changes in the hyperparameters can lead to large changes in training dynamics