PHP Singleton design pattern inheritance error

You made $instance private, which means it cannot be inherited. Change it to protected

protected static $instance;

The manual discusses the individual uses of visibility

http://php.net/manual/en/language.oop5.visibility.php

So it looks like the hangup is in your edit

$obj = Singleton::getInstance();
$obj_two = SingletonChild::getInstance();
var_dump($obj === $obj_two);   // false

This will never be true. getInstance gets an instance of the current class. Since they are different classes they are not the same. Doing a test like this is sloppy, however. I would never do a test on an object like this. What would make more sense is to make sure that you're getting an instance of Singleton which can be done very easily like so

if($obj_two instanceof Singleton) // true

Because the child inherits the parent, it is an instance of the parent

Inheriting Singleton class in PHP is difficult, event in PHP 7.0, but you can do this with some changes on your class to work.

first make your Singleton class to abstract

abstract class Singleton {

}

change your $instance variable to array $instance(s)

private $instances = [];

Now change getInstance() method like below

public static function getInstance() {
  if (!isset(self::$instances[static::class]) {
    self::$instances[static::class] = new static();
  }

  return self::$instances[static::class];
}

And change your test

remember now you can't call Singleton:: getInstance() due to abstract

class SingletonChild extends Singleton {
}

class SingletonChildTwo extends SingletonChild {
}

$obj = SingletonChild::getInstance();
$obj_two = SingletonChildTwo::getInstance();
var_dump($obj === SingletonChild::getInstance()); // true
var_dump($obj === $obj_two); // will -> false