No.
In a Rails system you've got Rails running, maybe across X number of instances with Unicorn or whatever, sitting behind nginx. If you want to have a long-running request or websocket you've typically used Go or Node or Elixir. If you need to store state between requests you may be using Redis. For background jobs and general async things you've probably got something like Sidekiq.
Erlang already has an http server that Elixir apps are using now, it's called Cowboy. It's not that Elixir is creating their own just because. One of the reasons nginx is often used for Rails apps is to handle static assets, because Rails just isn't as well-equipped to handle those. Cowboy seems to be fine with handling static assets, so a lot of people just take nginx out of the equation for an Elixir app.
Typically Rails apps have used something else like Node or Go or Elixir for long-running requests or websockets. Maybe that will change with ActionCable, maybe not. Either way, Elixir/Phoenix happen to be quite good at this already. Phoenix's equivalent to ActionCable is called Phoenix Channels and it's able to handle 2 million+ websocket connections on a single server, which is pretty cool I think.
Rails apps use redis to manage state between requests. Elixir apps just don't need something like this. Elixir doesn't have something different, it just happens to be easy to store state between requests already.
Rails apps may use Sidekiq or something to deal with background jobs or async operations. This is another case where Elixir doesn't have anything special to occupy that role in the stack, it just happens to be good at that already.
Most of these properties are things that Elixir inherited from Erlang, which has been around for years. I wish I had started learning Erlang a long time ago but unfortunately it just wasn't really on my radar. But really the only thing on this list which is a relatively recent thing is Phoenix Channels, and I don't think that's really a NIH thing. At least, certainly not any more than ActionCable is.