Where is the local backup located on Linux?

The extension will use the browser's local storage mechanism. For Chromium based browsers, this ends up stored in a profile specific folder normally in a location similar to $HOME/.config/google-chrome/Default/IndexedDB.

I'd like to know a little more about what your needs are in this case as I worry the local storage of the browser extension might not be the best way to achieve your goals. The local storage isn't explicitly designed for portability so you may find that exporting via the Linux app would produce more reliable backups.