ENH: Add ability to style axis names with `Styler`
- [ ] closes #48936
    - [x] apply_index
    - [ ] format_index
- [ ] [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
- [ ] 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.
- [ ] Added documentation to Table Visualization page

cc: @attack68 
I think this PR looks good but there are a few things to check off:

 - [ ] editing the styler guide documentation and any other locations for the list of CSS classes that are added in the HTML: `col_name` is a new one.
 - [ ] although we are re-using the `apply` and `applymap` methods with a small extension it is rather overkill to suggest that one would actually use a function to style these. It is probably easier just to supply a small list of styles to each name (of which there are probably a small number). In that case a simple documentation entry might help, something like:
```
styler.apply_index(lambda s: ["color: blue;", "color: red;"], names=True)
```
should suffice.
  - [ ] all other column CSS class names have `col` so I suggested sticking with the format.
  - [ ] needs a whats new and a version added for the argument, and a note about minor breaking change.
  - [ ] the copying mechanics are also changed. should review the docs there to see if it should be added.
  - [ ] the tests seem to be for multiindexes. Does it also work just in the regular index case? This pull request is stale because it has been open for thirty days with no activity. Please [update](https://pandas.pydata.org/pandas-docs/stable/development/contributing.html#updating-your-pull-request) and respond to this comment if you're still interested in working on this. > This pull request is stale because it has been open for thirty days with no activity. Please [update](https://pandas.pydata.org/pandas-docs/stable/development/contributing.html#updating-your-pull-request) and respond to this comment if you're still interested in working on this.

Still working on this Note to self on current progress:

- Completed implementation of https://github.com/pandas-dev/pandas/pull/48972#issuecomment-1272398597 except for checking the copy mechanics
- Need to generate and check documentation, specifically example output and style user guide (version added)
- Need to validate basic Index behavior and test completeness Thanks for the pull request, but it appears to have gone stale. If interested in continuing, please merge in the main branch, address any review comments and/or failing tests, and we can reopen.