There's nothing insidious or even new about this sense of the word "server." It's used for situations like virtualization, shared hosting, and even (unsurprisingly, given Discord's initial target market) multiplayer games. Decentralization and reproducibility are not parts of the definition.

the point is that they're all run together behind one interface, don't have any of the typical characteristics of servers (one or more of: hostnames, listening ports, separate authentication, ability to be installed on any machine).

Everything is hyperconverged

This is exactly how multiplayer games work.

That's not my experience of multiplayer games. If I run, say, a Minecraft server, I install the Minecraft server software on my computer and allow incoming connections to it. Sure I can get shared hosting, but I can still install it on my own computer/server if I want to. A few other games that I've seen or worked with in the past are the same way, if I run a server it's actually running as a separate installation on my own computer (it may be part of a centralised public server list, and it may be accessible only through a centralised proxy, but the actual server is still running on my computer). Discord is not like that, there are no separate servers, they're all hosted by a single company's infrastructure under their control (and I'm pretty sure there aren't any actual separate server software instances for each channel either, but that's mostly irrelevant at this point).

Sure I can get shared hosting, but I can still install it on my own computer/server if I want to.

I bet there are IRC clients on iOS and Android, but I don't think any of them are particularly well known except maybe outside a niche

Yes, I also hate that Discord is a completely centralized service, but things stagnated with both IM and IRC and when stagnation happens in open protocols what happens is that some centralized service will pick up the need and hog everything (and since everybody goes there, it'll be hard to convince people to even consider trying something else if an alternative comes up).

Aside from being niche, they are also horrible to use because they require a persistent connection which doesn't work out well on mobile devices. IRC cannot get around this without changing the protocol.

That's not how most online games work. Minecraft is an exception.

I still don't understand why open versions of something die, and then as soon as a centralised version comes along everyone's interested in it again.

IRC can be supplemented with logging and bouncers. Heck if enough people were interested in the protocol it would be easy to develop a set of extensions that allow downloading a backlog of conversation when connecting or refreshing (assuming that there's a logging server somewhere in the network).