It has nothing to do with the immutability of x. None is a singleton for which is and == return identical results, unless the type of x defines a weird __eq__ method. The name x is either bound or not bound to the same object as None. If not, the identity check will always be False, regardless of what other type it is. The result of the equality check is up to the type of x, but will be the same as the identity check for most sane classes.
Now the caveat is that while you can't override is, since that's an identity check done by the interpreter, you can override ==. Aside from speed and consistency, here's a toy example showing why x is None is preferable to x == None:
class Test:
    def __eq__(self, other):
        if other is None:
            return True
        return self is other
>>> Test() == None
True
>>> None == Test()
True
>>> Test() is None
False
>>> None is Test()
False
The reason that both Test () == None and None == Test() return the same result has to do with how comparisons are evaluated, and the fact that None does not define an explicit __eq__ method. Normally, the interpreter calls type(left).__eq__(left, right) on the operands. If the left operand's type does not define __eq__ or the method returns NotImplemented, it attempts type(right).__eq__(right, left). Only if both fail does it fail over to a built-in identity check. Since None doesn't define an equality check, it will always use the check defined by the other object before failing over to the identity check.