[Nadya]

I imagine it was trained on 'typical' anime-style art of black/bold character outlines and mostly flat colors.

A common solution to resizing anime characters is to create a colored vector of an image. The differences between these vectors and the original stills are minimal and usually 'satisfactory'. There is an entire scene of people who create these vectors and another scene of people who use the vectors to create wallpapers and other graphics. [0] Waifu2x can help replace the need to vector these images by increasing the quality of upscaling them.

This is the prevalent 'style' for anime - at least from the past 8-10 years or so. There are a few outliers and I imagine Waifu2x would work poorly on them. For example, I do not see it working well on a still from "The Garden of Words". [1]

[0] http://img04.deviantart.net/96b1/i/2015/105/8/d/oumae_kumiko...

[1] https://24framesps.files.wordpress.com/2014/11/the-garden-of...