BUG: join changes index type when doing join on empty dataframes
- [X] closes #52777
- [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).
- [X] Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) to new arguments/methods/functions.
- [X] Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature.

Looks good, but since 2.0.3 going out today, lets wait for this to go into 2.0.4 whatsnew for v2.0.4 isn't in the repo yet. Should I just move this to v2.1.0? Yeah let's just have this fix for 2.1 👍. I've moved it. Thanks again @topper-123 