Creating objects inside a C++ library (like std::cout)

I recently read somewhere that std::cout is an instance of the std::ostream class. I wish to implement a similar kind of thing. I make a class Animal and want to provide an instance of Animal class dog in the library itself like std::cout. I am not sure how to do it but here's a code snippet which hopefully will give you an idea of what I'm trying to achieve.

// lib.h

#ifndef LIB_H
#define LIB_H

#include <string>

class Animal {
public:
    Animal();
    std::string name;
};

Animal dog;
dog.name = "dog";
extern Animal dog;

#endif

---

// lib.cpp

#include "lib.h"

Animal::Animal() {}

Animal dog;
dog.name = "dog";

---

// main.cpp

#include "lib.h"
#include <iostream>

int main() {
    Animal my_dog = dog;
    std::cout << my_dog.name << std::endl;
    return 0;
}

This is the error I get when I try this code:

lib.cpp:6:1: error: ‘dog’ does not name a type
 dog.name = "dog";

This kind of code may seem silly, but I still am looking for ways to implement this approach and not its alternatives.