Disqus is using different web server to host the comments. Website owner like dramacool or kissasian or even blogs, EMBED the disqus' HTML into their websites....exactly like how people embedding IG, X, or FB posts.
You also can see same comments on different websites, if the owner embed the same "disqus comment section" everywhere on internet.
On another note, even though dramacool has announced the shut down, their associated websites are still accessible online. They just don't update with new videos as far as I observe. That's why people are still able to see your comments and therefore can reply or upvote.
