[backport 2.3.x] BUG: preserve (object) dtype in factorize (#60118) (#60174)

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

* BUG: preserve (object) dtype in factorize

* add fallback for float16

(cherry picked from commit 13926e5)

Author: jorisvandenbossche

pandas/core/base.py

@@ -48,6 +48,7 @@
 from pandas.core.dtypes.generic import (
     ABCDataFrame,
     ABCIndex,
+    ABCMultiIndex,
     ABCSeries,
 )
 from pandas.core.dtypes.missing import (
@@ -1198,13 +1199,18 @@ 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[