You can use closures to store state in Erlang, but generally state is kept by passing it through to further function calls.

I would say this makes Erlang mostly have explicit state, rather than saying Erlang is stateless.

Ets is implemented in C as global mutable state, but you could implement it in Erlang as a process per table (and a process to hold the list of tables) with no loss of functionality. You would send messages to fetch data or update data, etc. It's a performance and memory efficiency optimization to do it with C, of course.

It’s important to note Erlang’s global state tools are about controlled global state. The process registry is more elaborate than a simple naming scheme in say Python or Java but really not much harder to use. However that controlled registry allows the process registry to become multi-node with minimal or no code changes for the clients. It’s comparable in power to but far simpler than Consul or etcd for finding services. I view micro services as fairly equivalent to Erlang actors as they both reinforce that state encapsulation. The same idea goes for mnesia or ETS which are comparable in power to Redis.

Pure functional languages, aka Haskell or Idris or whatever, don't have this property. A closure closes over values, and thus is a value, no state mutation involved.

Of course, your point about complex mechanisms to reintroduce mutation into specific contexts stands (usually via monads) but those are a very different Kind of object then the functions.