The post you were replying to (unless it was edited after your reply) specifically mentioned a shopping cart cookie as one that could be classed as strictly necessary. There are other options but they have issues (tracking via query string or form values doesn't work well with multiple tabs open for instance). The cart ID can be the session ID too for as long as it is needed.
Of course they don't have to be stored, in fact they shouldn't be stored. They are session level naturally so belong in session level cookies not more permanent storage.
Also, while session tokens in cookies are usually fine to be defined as strictly essential for the main site, they are generally not for 3rd party cookies.
> localStorage/sessionStorage seems kind of like the same thing? Am I missing somehting?
No, those are more often used in equivalent ways to cookies though they don't do exactly the same job, extra logic is needed if your server-side needs to access the stored information. Cookie values are sent to the web server(s) with every request (except where certain flags are set), data in session/local storage needs to be explicitly read out and sent on in GET or POST parameters when needed.
