I'm not a network programmer or language designer by trade, so I expect to be missing something here, but I'll give it a go to learn where I'm wrong.

If you're using distinct interfaces for distinct states (as it seems you are in your latter examples) and your compiler is going to enforce them, then shadowing variables (as a language feature) lets you unassign them as you go along. In a language I'm more familiar with, which uses polymorphic types rather than interfaces:

let socket = socks5_socket () in
let socket = connect_unauthenticated socket ~proxy in
let socket = connect_tcp socket address in
enjoy socket ();

with signatures:

val sock5_socket : unit -> [> `Closed of Socket.t]
val connect_unauthenticated : [< `Closed of Socket.t] -> proxy:Address.t -> [> `Authenticated of Socket.t]
val connect_tcp -> [< `Authenticated of Socket.t] -> Address.t -> [> `Tcp_established of Socket.t]
val enjoy : [< `Tcp_established of Socket.t] -> unit -> unit

so that if you forget the connect_unauthenticated line (big danger of shadowing as you mutate), your compiler will correct you with a friendly but stern:

This expression has type [> `Closed of Socket.t] but an expression was expected of type [< `Authenticated of Socket.t].

Of course, shadowing without type-safety sounds like a nightmare, and I'm not claiming that any language you want to use actually supports it as syntax. But I occasionally appreciate it (given, of course, that I've got a type inspector readily keybound, to query what state my socket is in at this particular point).

And here’s an interesting observation: The API of the socket changes as you move from one state to another.

Anyway, this rant is addressed to programming language designers: What options do we have to support such mutating API at the moment. And can we do better?

It's called typestate. It's been tried, and it tends to be cumbersome.

You can do the linear typing thing in Rust. Have a hidden internal handle and API wrapper objects on top of it that get consumed on method calls and can return different wrappers holding the same handle. I took a shot at doing a toy implementation for the TCP case:

type internal_tcp_handle = usize;  // Hidden internal implementation

/// Initial closed state
#[derive(Debug)]
pub struct Tcp(internal_tcp_handle);

impl Tcp {
    pub fn connect_unauthenticated(self) -> Result<AuthTcp, Tcp> {
        // Consume current API wrapper,
        // return next state API wrapper with same handle.
        Ok(AuthTcp(self.0))
    }

    pub fn connect_password(self, _user: &str, pass: &str) -> Result<AuthTcp, Tcp> {
        // Can fail back to current state if password is empty.
        if pass.is_empty() { Err(self) } else { Ok(AuthTcp(self.0)) }
    }
}

/// Authenticated state.
#[derive(Debug)]
pub struct AuthTcp(internal_tcp_handle);

impl AuthTcp {
    pub fn connect_tcp(self, addr: &str) -> Result<TcpConnection, AuthTcp> {
        if addr.is_empty() { Err(self) } else { Ok(TcpConnection(self.0)) }
    }

    pub fn connect_udp(self, addr: &str) -> Result<UdpConnection, AuthTcp> {
        if addr.is_empty() { Err(self) } else { Ok(UdpConnection(self.0)) }
    }
}

#[derive(Debug)]
pub struct TcpConnection(internal_tcp_handle);

#[derive(Debug)]
pub struct UdpConnection(internal_tcp_handle);

fn main() {
    // Create unauthenticated TCP object.
    let tcp = Tcp(123);
    println!("Connection state: {:?}", tcp);

    // This would be a compiler error:
    // let tcp = tcp.connect_tcp("8.8.8.8").unwrap();
    // 'tcp' is bound to an API that doesn't support connect operations yet.

    // Rebind the stupid way, unwrap just runtime errors unless return is Ok.
    let tcp = tcp.connect_unauthenticated().unwrap();
    // Now 'tcp' is bound to the authenticated API, we can open connections.
    println!("Connection state: {:?}", tcp);

    // The runtime errory way is ugly, let's handle failure properly...
    if let Ok(tcp) = tcp.connect_tcp("8.8.8.8") {
        println!("Connection state: {:?}", tcp);
    } else {
        println!("Failed to connect to address!");
    }
    // TODO Now that we can use connected TCP methods on 'tcp',
    // implement those and write some actual network code...
}