"The article says that the data is basically "per-user","
Given that this is a table of who is "online", I don't think that's per-user in the sense that you are inferring. I infer that it's not a whole bunch of little local data that doesn't interact, it's a big global table of who is online and not online, constantly being heavily read from and written to in real time. Consider from the perspective of Bob's Erlang process that he wants to go offline and notify all of his currently-online friends that is is going offline. Bob's Erlang process doesn't have that data. Bob's Erlang process is going to get it from the Big Table of Who's Online. That table is the problem; it can't be stored in Bob's Erlang process.
I was at least imagining that the table could be partitioned into pieces pretty trivially (first X bits of the hash), but with Erlang's design, that implies an IPC just to ask some server process to give me the PID of the chunk I need to talk to, which itself is going to bottleneck. (In practice we'd probably cheat and use a NIF to do that, but that amounts to an admission that Erlang can't do this, so....)
At smaller scales you could try to live update Bob's local information as it changes, but this breaks down in all sorts of ways at scales far smaller than Discord, scales much closer to "a single mid-sized company".
"Another could be storing the data in mnesia, BEAM's internal mutable in-memory DB."
I have used mnesia for loads literally a ten-thousandth as small as this, if that (I could probably tack two more zeros on there), and it breaks down. It is an absolutely ludicrous idea that mnesia could handle what Discord is doing here. Last I knew the official Erlang community consensus was basically that mnesia really shouldn't be used for anything serious; my experience backed that up.
I think a non-trivial part of the reason why Erlang hasn't taken off is that its community still seems to exist in 2003, where it's a really incredible unique language that solves huge problems that nobody else does. In 2003, it rather has a point. But a lot of things have learned from Erlang, and incorporated its lessons into newer designs, and moved on.
See my other comment for what other runtimes have Erlang's advantages, but I'd invite you just to consider what we seem to basically agree on here; Erlang would be wildly slower and require a lot more hardware than Rust, the Rust code probably wasn't that hard to write, ... and the Rust code is way more likely to be correct than the Erlang code, too. I mean, what more "catching up to Elixir's inherent concurrency advantages" in this context than "did a job Elixir couldn't possibly do" do you want?