> Isn't Disqus a third-party service? Doesn't it use a cookie to know you have signed in?
Let's say Disqus JS code is embedded on a site A. Then it can set cookies for that domain. So when you enter your Disqus login and password on site A, it can send a request to Disqus server, obtain authorization token and save it in cookies for domain A. This way you will be recognized every time you visit site A.
This means that operators of site A may access those cookies too, but I don't see any problem here - it's their site anyway.
> If you put a Disqus comment form or Google Analytics in your website, wouldn't you need a cookie popup to comply with GDPR and similar regulations that regulate sending user data to third-parties?
Probably you need.
> Is 3P cookie supposed to be about the domain of the cookie?
3P cookie means that when site A includes content from site B (image, iframes) then the browser will send domain B's cookies with the request for that content. This means that if content from site B is included on 100 different websites, site B can track the user across them using cookies.
So when you sign into Disqus, it can recognize (and track your actions) you on any site using Disqus widget.
When 3P cookies are disabled, requests for content embedded from other sites like B, will be anonymous and without cookies. You will have to log into Disqus for every site where you want to leave a comment.