Bigger instances will indeed run multiple copies of the various components, it's pretty standard software in that regard.
Usually at first that will start by moving the PostgreSQL database to its own dedicated box, and then start adding additional backend boxes, possibly adding more caching in front so that the backend doesn't have to do as much work. Once the database is pegged, the next step is usually a write primary and one or more read secondaries. When that gets too much, you get into sharding so that you can spread the database load across multiple servers. I don't know much about PostgreSQL but I have to assume it's better than MySQL in that regard and I've seen a 1 TB MySQL database in the wild running just fine.
I think lemmy.world in general is hitting some scalability issues that they're working on. Keep in mind the software is fairly new and is just being truely tested at large scale, there's probably a ton of room for optimization. Also lemmy.world is still on 0.17 and apparently 0.18 changed the protocol a lot in a way that makes it scale much better, so when they complete that upgrade it'll probably run a lot better already.
The part that worries me about scalability in the long term is the push nature of ActivityPub. My server is already getting several POST requests to /inbox
per second already, which makes me wonder how that's gonna work if big instances have to push content updates to thousands of lemmy instances where most of the data probably isn't even seen. I was surprised it was a push system and not a pull system, as pull is much easier to scale and cache at the CDN level, and can be fetched on demand for people that only checks lemmy once in a while.
I need to start digging into Lemmy's code and get familiar with the internals, still only a couple days in with my private instance.