BUG: preserve _id in MultiIndex.copy(deep=False)

pandas/core/indexes/multi.py

@@ -1190,20 +1190,23 @@ def copy(
         This could be potentially expensive on large MultiIndex objects.
         """
         names = self._validate_names(name=name, names=names, deep=deep)
+        keep_id = not deep
         if levels is not None:
             warnings.warn(
                 "parameter levels is deprecated and will be removed in a future "
                 "version. Use the set_levels method instead.",
                 FutureWarning,
                 stacklevel=find_stack_level(),
             )
+            keep_id = False
         if codes is not None:
             warnings.warn(
                 "parameter codes is deprecated and will be removed in a future "
                 "version. Use the set_codes method instead.",
                 FutureWarning,
                 stacklevel=find_stack_level(),
             )
+            keep_id = False
 
         if deep:
             from copy import deepcopy
@@ -1225,6 +1228,8 @@ def copy(
         )
         new_index._cache = self._cache.copy()
         new_index._cache.pop("levels", None)  # GH32669
+        if keep_id:
+            new_index._id = self._id
 
         if dtype:
             warnings.warn(

pandas/tests/indexes/multi/test_copy.py

@@ -104,3 +104,15 @@ def test_copy_deprecated_parameters(deep, param_name, param_value):
         idx_copy = idx.copy(deep=deep, **{param_name: param_value})
 
     assert [list(i) for i in getattr(idx_copy, param_name)] == param_value
+
+
+def test_copy_deep_false_retains_id():
+    # GH#47878
+    idx = MultiIndex(
+        levels=[["foo", "bar"], ["fizz", "buzz"]],
+        codes=[[0, 0, 0, 1], [0, 0, 1, 1]],
+        names=["first", "second"],
+    )
+
+    res = idx.copy(deep=False)
+    assert res._id is idx._id