BUG: allow None as name in multi-index during join

doc/source/whatsnew/v3.0.0.rst

@@ -28,6 +28,7 @@ enhancement2
 
 Other enhancements
 ^^^^^^^^^^^^^^^^^^
+- :attr:`MultiIndex.names` containing ``None`` no longer throws ``AssertionError`` during join (:issue:`58721`)
 - :class:`pandas.api.typing.FrozenList` is available for typing the outputs of :attr:`MultiIndex.names`, :attr:`MultiIndex.codes` and :attr:`MultiIndex.levels` (:issue:`58237`)
 - :class:`pandas.api.typing.SASReader` is available for typing the output of :func:`read_sas` (:issue:`55689`)
 - :func:`DataFrame.to_excel` now raises an ``UserWarning`` when the character count in a cell exceeds Excel's limitation of 32767 characters (:issue:`56954`)

pandas/core/indexes/base.py

@@ -4516,8 +4516,8 @@ def _join_multi(self, other: Index, how: JoinHow):
         from pandas.core.reshape.merge import restore_dropped_levels_multijoin
 
         # figure out join names
-        self_names_list = list(com.not_none(*self.names))
-        other_names_list = list(com.not_none(*other.names))
+        self_names_list = list(self.names)
+        other_names_list = list(other.names)
         self_names_order = self_names_list.index
         other_names_order = other_names_list.index
         self_names = set(self_names_list)

pandas/tests/reshape/merge/test_join.py

@@ -1098,3 +1098,29 @@ def test_join_multiindex_categorical_output_index_dtype(how, values):
 
     result = df1.join(df2, how=how)
     tm.assert_frame_equal(result, expected)
+
+
+def test_join_multiindex_with_none_as_label():
+    # GH 58721
+    df1 = DataFrame(
+        {"A": [1]},
+        index=MultiIndex.from_tuples([(3, 3)], names=["X", None]),
+    )
+    df2 = DataFrame(
+        {"B": [2]},
+        index=MultiIndex.from_tuples([(3, 3)], names=[None, "X"]),
+    )
+
+    result12 = df1.join(df2)
+    expected12 = DataFrame(
+        {"A": [1], "B": [2]},
+        index=MultiIndex.from_tuples([(3, 3)], names=["X", None]),
+    )
+    tm.assert_frame_equal(result12, expected12)
+
+    result21 = df2.join(df1)
+    expected21 = DataFrame(
+        {"B": [2], "A": [1]},
+        index=MultiIndex.from_tuples([(3, 3)], names=[None, "X"]),
+    )
+    tm.assert_frame_equal(result21, expected21)