BUG: pd.concat fails when handling MultiIndex if repeated index exists

doc/source/whatsnew/v1.1.0.rst

@@ -555,6 +555,7 @@ Reshaping
 
 - Bug effecting all numeric and boolean reduction methods not returning subclassed data type. (:issue:`25596`)
 - Bug in :meth:`DataFrame.pivot_table` when only MultiIndexed columns is set (:issue:`17038`)
+- Bug in :func:`concat` will fail when handling MultiIndex if index is the same (:issue:`20565`)
 - Bug in :meth:`DataFrame.unstack` and :meth:`Series.unstack` can take tuple names in MultiIndexed data (:issue:`19966`)
 - Bug in :meth:`DataFrame.pivot_table` when ``margin`` is ``True`` and only ``column`` is defined (:issue:`31016`)
 - Fix incorrect error message in :meth:`DataFrame.pivot` when ``columns`` is set to ``None``. (:issue:`30924`)

pandas/tests/reshape/test_concat.py

@@ -1418,6 +1418,14 @@ def test_concat_rename_index(self):
         tm.assert_frame_equal(result, exp)
         assert result.index.names == exp.index.names
 
+    def test_concat_repeated_index(self):
+        # GH 20565
+        df = pd.DataFrame(np.random.randn(3, 2), columns=["A", "B"], index=["Z1"] * 3)
+
+        result = pd.concat([df, df], keys=["Key1", "Key2"], names=["KEY", "ID"])
+        expected = pd.Index(["Z1"], name="ID")
+        tm.assert_index_equal(result.index.levels[1], expected)
+
     def test_crossed_dtypes_weird_corner(self):
         columns = ["A", "B", "C", "D"]
         df1 = DataFrame(