In this context it means exactly that: staying on the client. And yes that means using cookies (or rather, local storage probably). So this is requiring consent no matter how you do it. But note that the system in the article also requires consent since it sends PII (the IP) over the wire and saves it on the server. It's reversible and not anonymous - so it's even worse than using a cookie without consent I'd say.

Yes if all you ever want to send is a unique visitor ID then there is no point in having a local hash, because you can just generate a random ID and use that to identify the user.

What I mean is that if you want to send multiple pieces of PII (such as an IP, a filename, a username,...) then the only way to do that safely is to send hash(salt+filename) for example, where the salt is not known to the server receiving the hash. The IP in the suggestion to use a locally stored hash here just represented "PII that should be sent anonymously" and not "A good way of identifying a unique system".