If you look closely enough at Haskell, you’ll realize that it also can have a lot of mutable state. Haskell just puts state into various monads and STM to make its functions pure. You can even spawn numerous isolated threads, all with their own state, and have them communicate with one another like you do in Elixir. It does provide a lot of structure and guarantees compared to your standard imperative language, but I assure you mutable state is still there.

To be clear, the Elm model of putting everything into a big tree and and transforming that at every user input is very unusual among even FP languages, and is not the model typically used in Haskell. This might be what you are thinking of.

Edit: I also want to point out that Elixir processes can be registered globally to act as singletons, but by default you can spawn any number of agents or other processes at runtime, meaning they are not singletons.