BUG (string dtype): comparison of string column to mixed object column fails #60228 (fixed)
#### Changes :
- Updated `comparison_op` function in `pandas/core/ops/array_ops.py`:
  - Added handling for comparisons between `string` and `object` dtypes.
  - Ensured `string` and `object` types are casted to a common type (`string`) before comparison.
- Added a test case to validate the fix:
  - Verified comparison works for `string` vs `object` with homogeneous and mixed types.
  - Verified behavior with PyArrow-based strings enabled (`pd.options.future.infer_string = True`).

--

- [x] closes #60228  (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).
- [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.

@TEARFEAR git workflow related comment: it seems you have picked up some unrelated commits. In general doing something like

```
git fetch upstream
git merge upstream/main
git push origin bug-update-60228
```

should fix that (assuming `origin` is your fork and `upstream` is the pandas-dev/pandas remote) (small ping to see if you have some time to update this PR) 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. @TEARFEAR would you have time to update this PR? (otherwise I am also happy to update it, as we would like to include this in the 2.3 release) 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. 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.