Earlier quoted context omitted.
It doesn't neccessarly have to be such a "ring" that you navigate from website #1 to website #2 and the order is always the same, and you cannot jump, or even do random browsing. I think initially the idea was mainly around the "ring" concept, but relatively quickly lots of different implementations did lots of different things, and we still called those "webrings" even though many weren't conceptually "rings" at all…
So part of it seems to be working together: I link to your site with the expectation that you link back to mine. Not a bad idea, also letting the site-creators coordinate who focuses on which sub-topic. But are there many examples of this in the real web?
I guess, but more collectively, so like "We both agree to share a list of websites we link to, and anyone who want to add their site to this list also agree to the same". It used to be popular in music, programming and web development blogs, that much I remember, but I can't say how popular it was at large, the web and the internet was pretty much mostly a geeky thing at the time (around 2000s).