diff --git a/modin/core/dataframe/pandas/dataframe/dataframe.py b/modin/core/dataframe/pandas/dataframe/dataframe.py
index 849e411164f..fe88066d7fe 100644
--- a/modin/core/dataframe/pandas/dataframe/dataframe.py
+++ b/modin/core/dataframe/pandas/dataframe/dataframe.py
@@ -581,9 +581,22 @@ def _set_columns(self, new_columns):
         if self.has_materialized_columns:
             # do not set new columns if they're identical to the previous ones
             if (
-                isinstance(new_columns, pandas.Index)
-                and self.columns.equals(new_columns)
-            ) or np.array_equal(self.columns.values, new_columns):
+                # `Index.equals()` doesn't compare metadata, thus we have to compare
+                # it manually. Here we process the simpliest and the most common case only
+                # (when index the index is a 'pandas.Index' dtype). Other cases are not that
+                # common and we can omit them
+                type(new_columns) in (pandas.Index, pandas.MultiIndex)
+                and (
+                    type(new_columns)
+                    is type(self.columns)  # noqa; here we need exact types comparison
+                )
+                and new_columns.name == self.columns.name
+                and new_columns.names == self.columns.names
+                and new_columns.equals(self.columns)
+            ) or (
+                not isinstance(new_columns, pandas.Index)
+                and np.array_equal(self.columns.values, new_columns)
+            ):
                 return
             new_columns = self._validate_set_axis(new_columns, self._columns_cache)
             if self.has_materialized_dtypes:
@@ -639,7 +652,6 @@ def _compute_axis_labels_and_lengths(self, axis: int, partitions=None):
         List of int
             Size of partitions alongside specified `axis`.
         """
-
         if partitions is None:
             partitions = self._partitions
         new_index, internal_idx = self._partition_mgr_cls.get_indices(axis, partitions)
diff --git a/modin/test/storage_formats/pandas/test_internals.py b/modin/test/storage_formats/pandas/test_internals.py
index f5af3845ee5..64c3063d082 100644
--- a/modin/test/storage_formats/pandas/test_internals.py
+++ b/modin/test/storage_formats/pandas/test_internals.py
@@ -1166,6 +1166,27 @@ def test_skip_set_columns():
     # Verifies that the new columns weren't set if they're equal to the previous ones
     assert not df._query_compiler._modin_frame._deferred_column
 
+    df = pd.DataFrame({"col1": [1, 2, 3], "col2": [3, 4, 5]})
+    df.columns = pandas.Index(["col1", "col2"], name="new name")
+    # Verifies that the new columns weren't set if they're equal to the previous ones
+    assert df.columns.name == "new name"
+
+    df = pd.DataFrame(
+        {("a", "col1"): [1, 2, 3], ("a", "col2"): [3, 4, 5], ("b", "col1"): [6, 7, 8]}
+    )
+    df.columns = df.columns.copy()
+    # Verifies that the new columns weren't set if they're equal to the previous ones
+    assert not df._query_compiler._modin_frame._deferred_column
+
+    df = pd.DataFrame(
+        {("a", "col1"): [1, 2, 3], ("a", "col2"): [3, 4, 5], ("b", "col1"): [6, 7, 8]}
+    )
+    new_cols = df.columns[::-1]
+    df.columns = new_cols
+    # Verifies that the new columns were successfully set in case they're actually new
+    assert df._query_compiler._modin_frame._deferred_column
+    assert df.columns.equals(new_cols)
+
     df = pd.DataFrame({"col1": [1, 2, 3], "col2": [3, 4, 5]})
     remove_axis_cache(df, axis=1)
     df.columns = ["col1", "col2"]