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

doc/source/whatsnew/v3.0.0.rst

@@ -654,6 +654,7 @@ Groupby/resample/rolling
 
 Reshaping
 ^^^^^^^^^
+- :attr:`MultiIndex.names` containing ``None`` no longer throws ``AssertionError`` during join (:issue:`58721`)
 - Bug in :func:`qcut` where values at the quantile boundaries could be incorrectly assigned (:issue:`59355`)
 - Bug in :meth:`DataFrame.join` inconsistently setting result index name (:issue:`55815`)
 - Bug in :meth:`DataFrame.merge` where merging on a column containing only ``NaN`` values resulted in an out-of-bounds array access (:issue:`59421`)

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)