object が同一かどうかはspl_object_hashで見ている http://php.net/manual/ja/function.spl-object-hash.php

1_static_property.php

<?php
class A {
    protected static $foo = null;
    function hoge() {
        if (is_null(self::$foo)) {
            self::$foo = new Inner();
        }
        return self::$foo;
    }
}

class B extends A {
    //protected static $foo = null;

    function hoge() {
        if (is_null(self::$foo)) {
            self::$foo = new Inner();
        }
        self::$foo->bar = 'hogehoge';
        return self::$foo;
    }
}

class Inner {
    public $bar = null;
}

$a = new A();
$b = new B();

echo spl_object_hash($a->hoge()) . "\n";
var_dump($a->hoge());
echo spl_object_hash($b->hoge()) . "\n";
var_dump($b->hoge());
echo spl_object_hash($a->hoge()) . "\n";
var_dump($a->hoge());

/*
0000000019cce013000000005e2dcd60
object(Inner)#3 (1) {
  ["bar"]=>
  NULL
}
0000000019cce013000000005e2dcd60
object(Inner)#3 (1) {
  ["bar"]=>
  string(8) "hogehoge"
}
0000000019cce013000000005e2dcd60
object(Inner)#3 (1) {
  ["bar"]=>
  string(8) "hogehoge"
}
*/

2_static_property_self.php

<?php
class A {
    protected static $foo = null;
    function hoge() {
        if (is_null(self::$foo)) {
            self::$foo = new Inner();
        }
        return self::$foo;
    }
}

class B extends A {
    protected static $foo = null;

    function hoge() {
        if (is_null(self::$foo)) {
            self::$foo = new Inner();
        }
        self::$foo->bar = 'hogehoge';
        return self::$foo;
    }
}

class Inner {
    public $bar = null;
}

$a = new A();
$b = new B();

echo spl_object_hash($a->hoge()) . "\n";
var_dump($a->hoge());
echo spl_object_hash($b->hoge()) . "\n";
var_dump($b->hoge());
echo spl_object_hash($a->hoge()) . "\n";
var_dump($a->hoge());

/*
0000000056b6bba5000000000c16550c
object(Inner)#3 (1) {
  ["bar"]=>
  NULL
}
0000000056b6bba2000000000c16550c
object(Inner)#4 (1) {
  ["bar"]=>
  string(8) "hogehoge"
}
0000000056b6bba5000000000c16550c
object(Inner)#3 (1) {
  ["bar"]=>
  NULL
}
*/

3_static_property_parent.php

<?php

class A {
    protected static $foo = null;
    function hoge() {
        if (is_null(self::$foo)) {
            self::$foo = new Inner();
        }
        return self::$foo;
    }
}

class Inner {
    public $bar = null;
}

class B extends A {
    protected static $foo = null;

    function hoge() {
        if (is_null(parent::$foo)) {
            parent::$foo = new Inner();
        }
        parent::$foo->bar = 'hogehoge';
        return parent::$foo;
    }
}

$a = new A();
$b = new B();

echo spl_object_hash($a->hoge()) . "\n";
var_dump($a->hoge());
echo spl_object_hash($b->hoge()) . "\n";
var_dump($b->hoge());
echo spl_object_hash($a->hoge()) . "\n";
var_dump($a->hoge());
/*
7b10dcffee174901c4dbe3b177cd7371
class Inner#3 (1) {
  public $bar =>
  NULL
}
7b10dcffee174901c4dbe3b177cd7371
class Inner#3 (1) {
  public $bar =>
  string(8) "hogehoge"
}
7b10dcffee174901c4dbe3b177cd7371
class Inner#3 (1) {
  public $bar =>
  string(8) "hogehoge"
}
*/

self の static 変数を定義すると、parent と self の同名変数が別のものとなるだけだったもよう。