I think we’re in a confusion of definitions here. The Discourse server-side code is (kind of*) stateless but the entire Discourse instance (web client, server side code, Redis, cached files, database) is not.
The other side of that medal is that - given that the server-side code is stateless - this setup can’t accomplish what the OP wants, since there is no place to store the information which URL and theme should be served. The setup you’re describing is actually what happens in a load-balancing setup where there are multiple web containers and a single database/Redis instance. It’s a single site.
* I’m saying “kind of” because there are a lot of caching layers in various places