diff --git a/doc/source/whatsnew/v0.24.0.rst b/doc/source/whatsnew/v0.24.0.rst
index f888648a9363e..a6171d39ae27f 100644
--- a/doc/source/whatsnew/v0.24.0.rst
+++ b/doc/source/whatsnew/v0.24.0.rst
@@ -1090,6 +1090,7 @@ Other API Changes
 - :meth:`Categorical.searchsorted` now raises a ``KeyError`` rather that a ``ValueError``, if a searched for key is not found in its categories (:issue:`23466`).
 - :meth:`Index.hasnans` and :meth:`Series.hasnans` now always return a python boolean. Previously, a python or a numpy boolean could be returned, depending on circumstances (:issue:`23294`).
 - The order of the arguments of :func:`DataFrame.to_html` and :func:`DataFrame.to_string` is rearranged to be consistent with each other. (:issue:`23614`)
+- :meth:`CategoricalIndex.reindex` now raises a ``ValueError`` if the target index is non-unique and not equal to the current index. It previously only raised if the target index was not of a categorical dtype (:issue:`23963`).
 
 .. _whatsnew_0240.deprecations:
 
@@ -1356,6 +1357,7 @@ Numeric
 - Bug in :meth:`Series.rpow` with object dtype ``NaN`` for ``1 ** NA`` instead of ``1`` (:issue:`22922`).
 - :meth:`Series.agg` can now handle numpy NaN-aware methods like :func:`numpy.nansum` (:issue:`19629`)
 - Bug in :meth:`Series.rank` and :meth:`DataFrame.rank` when ``pct=True`` and more than 2:sup:`24` rows are present resulted in percentages greater than 1.0 (:issue:`18271`)
+- Calls such as :meth:`DataFrame.round` with a non-unique :meth:`CategoricalIndex` now return expected data. Previously, data would be improperly duplicated (:issue:`21809`).
 
 Strings
 ^^^^^^^
diff --git a/pandas/core/indexes/category.py b/pandas/core/indexes/category.py
index 6b84e8deea493..1ad9f285b755f 100644
--- a/pandas/core/indexes/category.py
+++ b/pandas/core/indexes/category.py
@@ -532,12 +532,16 @@ def reindex(self, target, method=None, level=None, limit=None,
 
         target = ibase.ensure_index(target)
 
-        if not is_categorical_dtype(target) and not target.is_unique:
-            raise ValueError("cannot reindex with a non-unique indexer")
+        if self.equals(target):
+            indexer = None
+            missing = []
+        else:
+            if not target.is_unique:
+                raise ValueError("cannot reindex with a non-unique indexer")
 
-        indexer, missing = self.get_indexer_non_unique(np.array(target))
+            indexer, missing = self.get_indexer_non_unique(np.array(target))
 
-        if len(self.codes):
+        if len(self.codes) and indexer is not None:
             new_target = self.take(indexer)
         else:
             new_target = target
diff --git a/pandas/tests/frame/test_analytics.py b/pandas/tests/frame/test_analytics.py
index c9481fef4aa36..2bf2dd593184f 100644
--- a/pandas/tests/frame/test_analytics.py
+++ b/pandas/tests/frame/test_analytics.py
@@ -1805,6 +1805,21 @@ def test_built_in_round(self):
             {'col1': [1., 2., 3.], 'col2': [1., 2., 3.]})
         tm.assert_frame_equal(round(df), expected_rounded)
 
+    def test_round_nonunique_categorical(self):
+        # See GH21809
+        idx = pd.CategoricalIndex(['low'] * 3 + ['hi'] * 3)
+        df = pd.DataFrame(np.random.rand(6, 3), columns=list('abc'))
+
+        expected = df.round(3)
+        expected.index = idx
+
+        df_categorical = df.copy().set_index(idx)
+        assert df_categorical.shape == (6, 3)
+        result = df_categorical.round(3)
+        assert result.shape == (6, 3)
+
+        tm.assert_frame_equal(result, expected)
+
     def test_pct_change(self):
         # GH 11150
         pnl = DataFrame([np.arange(0, 40, 10), np.arange(0, 40, 10), np.arange(
diff --git a/pandas/tests/indexes/test_category.py b/pandas/tests/indexes/test_category.py
index 6c5a70d76e3b5..937e5e5a6af51 100644
--- a/pandas/tests/indexes/test_category.py
+++ b/pandas/tests/indexes/test_category.py
@@ -540,6 +540,17 @@ def test_reindex_dtype(self):
         tm.assert_numpy_array_equal(indexer,
                                     np.array([0, 3, 2], dtype=np.intp))
 
+    def test_reindex_duplicate_target(self):
+        # See GH23963
+        c = CategoricalIndex(['a', 'b', 'c', 'a'],
+                             categories=['a', 'b', 'c', 'd'])
+        with pytest.raises(ValueError, match='non-unique indexer'):
+            c.reindex(['a', 'a', 'c'])
+
+        with pytest.raises(ValueError, match='non-unique indexer'):
+            c.reindex(CategoricalIndex(['a', 'a', 'c'],
+                                       categories=['a', 'b', 'c', 'd']))
+
     def test_reindex_empty_index(self):
         # See GH16770
         c = CategoricalIndex([])