BUG: Fix date_format and datetime_format arguments to ExcelWriter being ignored
- [X] closes #44284
- [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.

Notes:
* The fix is as suggested in the [latest comment](https://github.com/pandas-dev/pandas/issues/44284#issuecomment-992227644) on the issue.
* I applied the fix to all the writers, not just Openpyxl. This was following the pattern of changes made in the PR where the bug was introduced: https://github.com/pandas-dev/pandas/pull/40430
* The fix is very straightforward (2-line change in each of 3 files); the challenge was adding tests for it, since to properly test it we need to inspect the formatting info on the Excel file. Crucially, there is not a common API for this within pandas across the different engines, so initially adding the test into the `TestExcelWriter` class, which is parametrized over the different writers, gave some quite awkward code branching between the different APIs for the different engines. So I decided to move the tests to the test files for each individual engine, though this leaves some duplication between them.

Potential outstanding issues:
* Moving the test from `TextExcelWriter` to the individual test files meant losing the `openpyxl-.xlsm` parameter, since there didn't seem to be a file where this extension was being tested.
* Even with the arguments being passed through, `ODSWriter` does not seem to respect them, and this may be a limitation of the ODF specification. This felt like it was out-of-scope for this bug. Consequently, there was not a test added for the ODF engine.
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.