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In Japanese, these outfits are called "zenshin tights" (全身タイツ), with "zenshin" meaning "full body". The tights are usually lycra or sometimes latex, and they also have a following in other countries.
Zenshin tights are closely connected to sexuality, specifically "zenshin fechi" (全身フェチ) or "zenshin fetishism". In the early 1990s, a collection of adult zenshin photos called Body Discipline were released in Japan, and helped spark interest in the outfits. Online in Japan, people created zenshin communities. Those in the subculture even started holding offline parties for those who wore the outfits.
In Japan, the outfits are certainly not mainstream, and they're not always sexualized. Sometimes they are worn for laughs. A few years back, there were even a few Japanese comedians, such as Moeyan, who have performed while wearing zenshin tights.
Previously, Kotaku posted a skinsuit featuring Totoro from the classic anime My Neighbor Totoro. The roots of these outfits are the costumes worn by superheroes. That's perhaps one reason why the above gallery is packed full of Japanese superheroes like the Power Rangers or Ultraman. The other reason is that people who buy these outfits are interested in dressing up like superheroes.
But as the Totoro outfit shows, there is interest in dressing up as anime characters as well as video game ones, too. It's not exactly certain if these costumes are licensed. It is certain that you're not likely to forget them anytime soon. Or ever.