BUG: Fix Series comparison fails when index dtypes differ (object vs string) (#61099)
- [x] closes #61099
- [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.
- [x] Added an entry in the latest `doc/source/whatsnew/v3.0.0.rst` file if fixing a bug or adding a new feature.
@MayurKishorKumar - looks like your environment got added to your commit. Oh shoot, my bad! I’ll fix that right away. Thanks for catching it! Hi @rhshadrach 👋

I’m working on fixing [[#61099](https://github.com/pandas-dev/pandas/issues/61099)] and ran into a failure in `test_mixed_col_index_dtype`.

My fix updates `Index.equals` so that `StringDtype` and `object` dtypes are treated as equivalent when comparing column indexes. As a result, this test now fails because `result.columns.dtype` becomes `"string"` while `expected.columns.dtype` remains `object`.

There are two options I’m considering:

1. **Update the test** to explicitly cast `expected.columns` to `"string"` when `using_infer_string=True`, so it reflects the result.
2. **Adjust internal logic** so the result stays `object`, but that might go against the spirit of treating string/object as equal.

Would updating the test be acceptable in this case?

Thanks!
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