BUG: make JSONTableWriter fail if no index.name and 'index' in columns

This commit is itended to fix a bug GH pandas-dev#58925. If index.name is empty it
will use set_default_names inside __init__ to make check on overlapping
names fail. Otherwise it's done during schema creation and not reflected
on the dataframe itself which creates inconsistency between the data and
its schema.

Author: taranarmo

pandas/io/json/_json.py

@@ -60,6 +60,7 @@
 from pandas.io.json._table_schema import (
     build_table_schema,
     parse_table_schema,
+    set_default_names,
 )
 from pandas.io.parsers.readers import validate_integer
 
@@ -352,7 +353,10 @@ def __init__(
             )
             raise ValueError(msg)
 
+        obj = obj.copy()
         self.schema = build_table_schema(obj, index=self.index)
+        if self.index:
+            obj = set_default_names(obj)
 
         # NotImplemented on a column MultiIndex
         if obj.ndim == 2 and isinstance(obj.columns, MultiIndex):
@@ -369,7 +373,6 @@ def __init__(
             msg = "Overlapping names between the index and columns"
             raise ValueError(msg)
 
-        obj = obj.copy()
         timedeltas = obj.select_dtypes(include=["timedelta"]).columns
         if len(timedeltas):
             obj[timedeltas] = obj[timedeltas].map(lambda x: x.isoformat())