BUG: Index.join with different names doesn't return None name

doc/source/whatsnew/v3.0.0.rst

@@ -116,9 +116,6 @@ Performance improvements
 
 Bug fixes
 ~~~~~~~~~
-- Fixed bug in :meth:`DataFrame.join` inconsistently setting result index name (:issue:`55815`)
-- Fixed bug in :meth:`Series.diff` allowing non-integer values for the ``periods`` argument. (:issue:`56607`)
--
 
 Categorical
 ^^^^^^^^^^^
@@ -197,8 +194,8 @@ Groupby/resample/rolling
 
 Reshaping
 ^^^^^^^^^
--
--
+- Bug in :meth:`DataFrame.join` inconsistently setting result index name (:issue:`55815`)
+- Bug in :meth:`Index.join` where different :class:`Index` with different ``name`` s would not return an :class:`Index` with ``name=None`` (:issue:`57065`)
 
 Sparse
 ^^^^^^
@@ -218,6 +215,8 @@ Styler
 Other
 ^^^^^
 - Bug in :meth:`DataFrame.where` where using a non-bool type array in the function would return a ``ValueError`` instead of a ``TypeError`` (:issue:`56330`)
+- Bug in :meth:`Series.diff` allowing non-integer values for the ``periods`` argument. (:issue:`56607`)
+
 
 
 .. ***DO NOT USE THIS SECTION***

pandas/core/indexes/base.py

@@ -4834,6 +4834,8 @@ def _join_non_unique(
         else:
             join_index = self.take(left_idx)
 
+        join_index.name = ops.get_op_result_name(self, other)
+
         if how == "outer":
             mask = left_idx == -1
             if mask.any():

pandas/tests/indexes/numeric/test_join.py

@@ -366,3 +366,12 @@ def test_join_outer(self, index_large):
         tm.assert_index_equal(res, eres)
         tm.assert_numpy_array_equal(lidx, elidx)
         tm.assert_numpy_array_equal(ridx, eridx)
+
+
+def test_join_differing_names_none():
+    # GH 57065
+    idx1 = Index([1, 1, 2, 3, 4, 4], name="a")
+    idx2 = Index([2, 5, 3, 4], name="b")
+    result = idx1.join(idx2)
+    expected = Index([1, 1, 2, 3, 4, 4])
+    tm.assert_index_equal(result, expected)