A useful distinction
The Web Storage API distinguishes storage associated with an origin that can persist across sessions from storage associated with a page session. That distinction does not mean a game necessarily uses either mechanism or provides a reliable account backup.
Put it into practice
Read the game’s own description of saving. In a disposable prototype, record which mechanism is used and what your test observed after a reload or new session. Do not clear a real player’s data to experiment, and do not promise indefinite persistence from the API name alone.