Multisite won’t work guys,
and this won’t work either
Discourse is not stateless, and all the information is in the database. So if you run multiple instances on top of a single database then they would automatically look the same and changes in one would immediately reflect in the other.
This would “just” be a matter of switching themes. Maybe leverage the existing theme preview logic and have a plugin just look at the hostname instead of the URL parameter.
Hosting the same forum under two URL’s might be the hardest part here, especially with respect to SEO. It would get you into all kinds of duplicate content and canonical URL trouble.