It is also referred to as a tasuki (襷, another kimono accessory), or more narrowly as obigami (帯紙, "belt paper").
[2] The terms obi and tasuki are also used for a strip that is looped over one side (usually on the left) or folded over the top of (vinyl) LP albums released in Japan, and folded over the left side of music CDs, video games, LaserDiscs, or DVDs.
In this particular context, those obi in cardboard are commonly called spine cards in English, particularly by collectors of Japanese artifacts.
With the exceptions of books where belly-bands are used to add marketing claims, obi were unique to Japan and are used to provide the title of the product, track listings (if applicable), price, catalog number and information on related releases in Japanese.
Nowadays, Japanese publishers also release internationally some collector's edition of video games with their obi.