operator.index with custom class instance

This actually appears to be a deep-rooted issue in cpython. If you look at the source code for operator.py, you can see the definition of index:

def index(a):
    "Same as a.__index__()."
    return a.__index__()

So...why is it not equivalent? It's literally calling __index__. Well, at the bottom of the source, there's the culprit:

try:
    from _operator import *
except ImportError:
    pass
else:
    from _operator import __doc__

It's overwriting the definitions with a native _operator module. In fact, if you comment this out (either by modifying the actual library or making your own fake operator.py* and importing that), it works. So, we can find the source code for the native _operator library, and look at the related part:

static PyObject *
_operator_index(PyObject *module, PyObject *a)
{
    return PyNumber_Index(a);
}

So, it's a wrapper around the PyNumber_Index function. PyNumber_Index is a wrapper around _PyNumber_Index, so we can look at that:

PyObject *
_PyNumber_Index(PyObject *item)
{
    PyObject *result = NULL;
    if (item == NULL) {
        return null_error();
    }

    if (PyLong_Check(item)) {
        Py_INCREF(item);
        return item;
    }
    if (!_PyIndex_Check(item)) {
        PyErr_Format(PyExc_TypeError,
                     "'%.200s' object cannot be interpreted "
                     "as an integer", Py_TYPE(item)->tp_name);
        return NULL;
    }
    result = Py_TYPE(item)->tp_as_number->nb_index(item);
    if (!result || PyLong_CheckExact(result))
        return result;
    if (!PyLong_Check(result)) {
        PyErr_Format(PyExc_TypeError,
                     "__index__ returned non-int (type %.200s)",
                     Py_TYPE(result)->tp_name);
        Py_DECREF(result);
        return NULL;
    }
    /* Issue #17576: warn if 'result' not of exact type int. */
    if (PyErr_WarnFormat(PyExc_DeprecationWarning, 1,
            "__index__ returned non-int (type %.200s).  "
            "The ability to return an instance of a strict subclass of int "
            "is deprecated, and may be removed in a future version of Python.",
            Py_TYPE(result)->tp_name)) {
        Py_DECREF(result);
        return NULL;
    }
    return result;
}

PyObject *
PyNumber_Index(PyObject *item)
{
    PyObject *result = _PyNumber_Index(item);
    if (result != NULL && !PyLong_CheckExact(result)) {
        Py_SETREF(result, _PyLong_Copy((PyLongObject *)result));
    }
    return result;
}

You can see before it even calls nb_index (the C name for __index__), it calls PyLong_Check on the argument, and if it's true, it just returns the item with no modification. PyLong_Check is a macro that checks for long subtyping (int in python is a PyLong):

#define PyLong_Check(op) \
        PyType_FastSubclass(Py_TYPE(op), Py_TPFLAGS_LONG_SUBCLASS)
#define PyLong_CheckExact(op) Py_IS_TYPE(op, &PyLong_Type)

So, basically, the takeaway is that for whatever reason, probably for speed, int subclasses don't get their __index__ method called, and instead just get _PyLong_Copy'd to the resulting return value, but only in the native _operator module, and not in the non-native operator.py. This conflict of implementation as well as inconsistency in documentation leads me to believe that this is an issue, either in the documentation or the implementation, and you may want to raise it as one.

It's likely a documentation and not an implementation issue, as cpython has a habit of sacrificing correctness for speed: (nan,) == (nan,) but nan != nan.

* You may have to name it something like fake_operator.py then import it with import fake_operator as operator

This is because your type is an int subclass. __index__ will not be used because the instance is already an integer. That much is by design, and unlikely to be considered a bug in CPython. PyPy behaves the same.

In _operator.c:

static PyObject *
_operator_index(PyObject *module, PyObject *a)
/*[clinic end generated code: output=d972b0764ac305fc input=6f54d50ea64a579c]*/
{
    return PyNumber_Index(a);
}

Note that operator.py Python code is not used generally, this code is only a fallback in the case that compiled _operator module is not available. That explains why the result a.__index__() differs.

In abstract.c, cropped after the relevant PyLong_Check part:

/* Return an exact Python int from the object item.
   Raise TypeError if the result is not an int
   or if the object cannot be interpreted as an index.
*/
PyObject *
PyNumber_Index(PyObject *item)
{
    PyObject *result = _PyNumber_Index(item);
    if (result != NULL && !PyLong_CheckExact(result)) {
        Py_SETREF(result, _PyLong_Copy((PyLongObject *)result));
    }
    return result;
}

...

/* Return a Python int from the object item.
   Can return an instance of int subclass.
   Raise TypeError if the result is not an int
   or if the object cannot be interpreted as an index.
*/
PyObject *
_PyNumber_Index(PyObject *item)
{
    PyObject *result = NULL;
    if (item == NULL) {
        return null_error();
    }

    if (PyLong_Check(item)) {
        Py_INCREF(item);
        return item;     /* <---- short-circuited here */
    }
    ...
}

The documentation for operator.index is inaccurate, so this may be considered a minor documentation issue:

>>> import operator
>>> operator.index.__doc__
'Same as a.__index__()'

So, why isn't __index__ considered for integers? The probable answer is found in PEP 357, under the discussion section titled Speed:

Implementation should not slow down Python because integers and long integers used as indexes will complete in the same number of instructions. The only change will be that what used to generate an error will now be acceptable.

We do not want to slow down the most common case for slicing with integers, having to check for an nb_index slot every time.