BUG: assert_frame_equal still checks category dtypes even when asked not to check index type
- [x] closes #52126
- [ ] [Tests added and passed]
- [x] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [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 pretty good. Could use a whatsnew note in `v2.0.1.rst` @mroeschke done Thanks @jayendra-patil33 