Fix module type hint (#5469)

* Fix module type hint

"module" is not a valid python value.
The correct type hint for a module object is "types.ModuleType" which has existed since at least Python 2.6

* Added module type hint test

* style: pre-commit fixes

* Remove doc function

* Fixed type hint

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>

include/pybind11/pybind11.h

@@ -1322,7 +1322,7 @@ PYBIND11_NAMESPACE_BEGIN(detail)
 
 template <>
 struct handle_type_name<module_> {
-    static constexpr auto name = const_name("module");
+    static constexpr auto name = const_name("types.ModuleType");
 };
 
 PYBIND11_NAMESPACE_END(detail)

tests/test_modules.py

@@ -81,6 +81,13 @@ def test_pydoc():
     assert pydoc.text.docmodule(pybind11_tests)
 
 
+def test_module_handle_type_name():
+    assert (
+        m.def_submodule.__doc__
+        == "def_submodule(arg0: types.ModuleType, arg1: str) -> types.ModuleType\n"
+    )
+
+
 def test_duplicate_registration():
     """Registering two things with the same name"""