diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 88481160a2860..40215ea87f978 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -3312,7 +3312,8 @@ def _get_indexer(
             tolerance = self._convert_tolerance(tolerance, target)
 
         if not is_dtype_equal(self.dtype, target.dtype):
-            dtype = find_common_type([self.dtype, target.dtype])
+            dtype = self._find_common_type_compat(target)
+
             this = self.astype(dtype, copy=False)
             target = target.astype(dtype, copy=False)
             return this.get_indexer(
@@ -5075,15 +5076,7 @@ def get_indexer_non_unique(self, target):
         if not is_dtype_equal(self.dtype, target.dtype):
             # TODO: if object, could use infer_dtype to preempt costly
             #  conversion if still non-comparable?
-            dtype = find_common_type([self.dtype, target.dtype])
-            if (
-                dtype.kind in ["i", "u"]
-                and is_categorical_dtype(target.dtype)
-                and target.hasnans
-            ):
-                # FIXME: find_common_type incorrect with Categorical GH#38240
-                # FIXME: some cases where float64 cast can be lossy?
-                dtype = np.dtype(np.float64)
+            dtype = self._find_common_type_compat(target)
 
             this = self.astype(dtype, copy=False)
             that = target.astype(dtype, copy=False)
@@ -5196,6 +5189,20 @@ def _maybe_promote(self, other: Index):
 
         return self, other
 
+    def _find_common_type_compat(self, target: Index) -> DtypeObj:
+        """
+        Implementation of find_common_type that adjusts for Index-specific
+        special cases.
+        """
+        dtype = find_common_type([self.dtype, target.dtype])
+        if dtype.kind in ["i", "u"]:
+            # TODO: what about reversed with self being categorical?
+            if is_categorical_dtype(target.dtype) and target.hasnans:
+                # FIXME: find_common_type incorrect with Categorical GH#38240
+                # FIXME: some cases where float64 cast can be lossy?
+                dtype = np.dtype(np.float64)
+        return dtype
+
     @final
     def _should_compare(self, other: Index) -> bool:
         """
diff --git a/pandas/tests/arrays/categorical/test_indexing.py b/pandas/tests/arrays/categorical/test_indexing.py
index 6068166cb8618..5b31776301f7b 100644
--- a/pandas/tests/arrays/categorical/test_indexing.py
+++ b/pandas/tests/arrays/categorical/test_indexing.py
@@ -212,17 +212,25 @@ def test_categories_assignments_wrong_length_raises(self, new_categories):
     # Combinations of missing/unique
     @pytest.mark.parametrize("key_values", [[1, 2], [1, 5], [1, 1], [5, 5]])
     @pytest.mark.parametrize("key_class", [Categorical, CategoricalIndex])
-    def test_get_indexer_non_unique(self, idx_values, key_values, key_class):
+    @pytest.mark.parametrize("dtype", [None, "category", "key"])
+    def test_get_indexer_non_unique(self, idx_values, key_values, key_class, dtype):
         # GH 21448
         key = key_class(key_values, categories=range(1, 5))
+
+        if dtype == "key":
+            dtype = key.dtype
+
         # Test for flat index and CategoricalIndex with same/different cats:
-        for dtype in [None, "category", key.dtype]:
-            idx = Index(idx_values, dtype=dtype)
-            expected, exp_miss = idx.get_indexer_non_unique(key_values)
-            result, res_miss = idx.get_indexer_non_unique(key)
+        idx = Index(idx_values, dtype=dtype)
+        expected, exp_miss = idx.get_indexer_non_unique(key_values)
+        result, res_miss = idx.get_indexer_non_unique(key)
+
+        tm.assert_numpy_array_equal(expected, result)
+        tm.assert_numpy_array_equal(exp_miss, res_miss)
 
-            tm.assert_numpy_array_equal(expected, result)
-            tm.assert_numpy_array_equal(exp_miss, res_miss)
+        exp_unique = idx.unique().get_indexer(key_values)
+        res_unique = idx.unique().get_indexer(key)
+        tm.assert_numpy_array_equal(res_unique, exp_unique)
 
     def test_where_unobserved_nan(self):
         ser = Series(Categorical(["a", "b"]))