“Rattan” is the name used to describe about 600 species of climbing palm plants native to the tropical forests of Southeast Asia. Rattan vines, while extremely pliable and bendable, are still considers solid wood, which makes them strong and durable, especially when weaved together.
You may have also heard the term “wicker”. “Wicker” is not a material, but a method of weaving. Some wicker furniture use bamboo and synthetic fibers, but rattan continues to be the most popular kind of wicker furniture. While rattan furniture has been ubiquitous in Asia for centuries, it became popular in the West in the Victorian era as a way of giving rooms in their home an “exotic” feel.
This specific style chair in Animal Crossing is called a “barrel chair”