TST: Adds test coverage for pd.read_json index type
- [x] closes #28558 
- [x] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests) if fixing a bug or adding a new feature
- [x] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [ ] Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) to new arguments/methods/functions.
- [ ] Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature.

## Description: 
@mroeschke mentioned that the merged fix for issue [#28558](https://github.com/pandas-dev/pandas/issues/28558) could use a [test](https://github.com/pandas-dev/pandas/issues/28558#issuecomment-1528187186). This PR adds the missing test for index type equality between the `DataFrame` and `read_json` functions.

Updated with your suggestions @mroeschke  Updated with your latest suggestion @mroeschke. Let me know if there is anything else.  Thanks @mcgeestocks 