EHN: `df.to_latex(escape=True)` also escape index names
- [ ] closes #57362 (Replace xxxx with the GitHub issue number)
- [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.

Stage 3. Part of an multi-stages effort: https://github.com/pandas-dev/pandas/pull/57880#issuecomment-2003636401
Part 2 has a issue at https://github.com/pandas-dev/pandas/issues/59324
But in the process of implementing, I realized the current implementation of `df.to_latex(escape)` does not go through `styler.to_latex` but to call `styler.format_index`. This implementation follows the same flow. So maybe part 2 is not really relevant.

cc @attack68 as the codeowner of styler Thanks @quangngd 