BUG: preserve (object) dtype in factorize (#60118)

* BUG: preserve (object) dtype in factorize

* add fallback for float16

pandas/core/base.py

@@ -44,6 +44,7 @@
 from pandas.core.dtypes.generic import (
     ABCDataFrame,
     ABCIndex,
+    ABCMultiIndex,
     ABCSeries,
 )
 from pandas.core.dtypes.missing import (
@@ -1287,13 +1288,18 @@ def factorize(
         if uniques.dtype == np.float16:
             uniques = uniques.astype(np.float32)
 
-        if isinstance(self, ABCIndex):
-            # preserve e.g. MultiIndex
+        if isinstance(self, ABCMultiIndex):
+            # preserve MultiIndex
             uniques = self._constructor(uniques)
         else:
             from pandas import Index
 
-            uniques = Index(uniques)
+            try:
+                uniques = Index(uniques, dtype=self.dtype)
+            except NotImplementedError:
+                # not all dtypes are supported in Index that are allowed for Series
+                # e.g. float16 or bytes
+                uniques = Index(uniques)
         return codes, uniques
 
     _shared_docs["searchsorted"] = """

pandas/tests/test_algos.py

@@ -65,7 +65,6 @@ def test_factorize_complex(self):
         expected_uniques = np.array([(1 + 0j), (2 + 0j), (2 + 1j)], dtype=complex)
         tm.assert_numpy_array_equal(uniques, expected_uniques)
 
-    @pytest.mark.xfail(using_string_dtype(), reason="TODO(infer_string)", strict=False)
     def test_factorize(self, index_or_series_obj, sort):
         obj = index_or_series_obj
         result_codes, result_uniques = obj.factorize(sort=sort)