BUG: Fix TypeError in assert_index_equal when comparing CategoricalIndex with check_categorical=True and exact=False

pandas/_testing/asserters.py

@@ -322,7 +322,17 @@ def _check_types(left, right, obj: str = "Index") -> None:
     # skip exact index checking when `check_categorical` is False
     elif check_exact and check_categorical:
         if not left.equals(right):
-            mismatch = left._values != right._values
+            try:
+                mismatch = left._values != right._values
+            except TypeError:
+                if hasattr(left, "_internal_get_values") and hasattr(
+                    right, "_internal_get_values"
+                ):
+                    mismatch = (
+                        left._internal_get_values() != right._internal_get_values()
+                    )
+                else:
+                    mismatch = left.values != right.values
 
             if not isinstance(mismatch, np.ndarray):
                 mismatch = cast("ExtensionArray", mismatch).fillna(True)

pandas/tests/util/test_assert_index_equal.py

@@ -317,3 +317,12 @@ def test_assert_multi_index_dtype_check_categorical(check_categorical):
             tm.assert_index_equal(idx1, idx2, check_categorical=check_categorical)
     else:
         tm.assert_index_equal(idx1, idx2, check_categorical=check_categorical)
+
+
+def test_assert_index_equal_categorical_mismatch_categories():
+    # GH#61941
+    ci1 = CategoricalIndex(["a", "b", "c"], categories=["a", "b", "c"], ordered=False)
+    ci2 = CategoricalIndex(["a", "x", "c"], categories=["a", "b", "c"], ordered=False)
+
+    with pytest.raises(AssertionError, match="Index are different"):
+        tm.assert_index_equal(ci1, ci2, check_exact=False, check_categorical=True)