Add unit test and (GH#61941)

Author: Aniketsy

pandas/_testing/asserters.py

@@ -323,14 +323,9 @@ def _check_types(left, right, obj: str = "Index") -> None:
     elif check_exact and check_categorical:
         if not left.equals(right):
             try:
-                mismatch = ( 
-                    left._internal_get_values() != right._internal_get_values()
-                )
-            except TypeError as e:
-                raise AssertionError(
-                    f"{obj} cannot be compared due to incompatible"
-                    f"categorical types.\n{e}"
-                ) from e
+                mismatch = left._values != right._values
+            except TypeError :
+                mismatch = left._internal_get_values() != right._internal_get_values()    
 
             if not isinstance(mismatch, np.ndarray):
                 mismatch = cast("ExtensionArray", mismatch).fillna(True)

pandas/tests/util/test_assert_index_equal.py

@@ -320,8 +320,8 @@ def test_assert_multi_index_dtype_check_categorical(check_categorical):
 
 def test_assert_index_equal_categorical_mismatch_categories():
     # GH#61941
-    left = CategoricalIndex(["a", "b"], categories=["a", "b"])
-    right = CategoricalIndex(["a", "b"], categories=["b", "a"])
+    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="cannot be compared due to incompatible"):
-        tm.assert_index_equal(left, right, check_exact=True, check_categorical=True)
+    with pytest.raises(AssertionError, match="Index are different"):
+        tm.assert_index_equal(ci1, ci2, check_exact=False, check_categorical=True)