Yes, this forum is built with Discourse.
And no, you cannot host it here. Discourse is quite a complex beast, and requires Ruby, PostgreSQL and Redis to run. The official way to run it is with Docker.
Maybe you can find a web hosting service that provides the necessary technologies, and maybe with enough effort you can make Discourse work on it. But just getting your own server and running the Docker stack is easier.
If you’re looking for an alternative that you can host with us, I suggest you try Flarum. Flarum isn’t as big and mature as Discourse, but the design ideas are similar. And Flarum only needs PHP and MySQL.