Refer to the official documentation here on available storage options that web apps can use.
Local storage persists across browser sessions. It is stored in the user's cache and remains there until it is cleaned up by the user or via the web app.
If the data is not persisting across browser sessions then it is using session storage, not local storage, or the web app is wiping the data. This is common for cached data the site might use. I guess it could also happen if you were running an in private window or something similar as well.