DOC: fix doctests for pandas/core/strings/accessor.py for new string dtype
- [ ] ~closes #xxxx (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.~

Part of #61886
cc @jorisvandenbossche 

I noticed that some of the docstrings highlight differences between returning `NaN` and `False` for boolean operations (e.g., [here](https://github.com/arthurlw/pandas/blob/1613ce992f0ab4a34ab4b0db1a421efd63b2fd54/pandas/core/strings/accessor.py#L1296)), which are now outdated. These should be updated, though it might be better to open a separate issue for that.

Other files may have similar cases, though I haven't done a full check yet. > I noticed that some of the docstrings highlight differences between returning `NaN` and `False` for boolean operations (e.g., [here](https://github.com/arthurlw/pandas/blob/1613ce992f0ab4a34ab4b0db1a421efd63b2fd54/pandas/core/strings/accessor.py#L1296)), which are now outdated. These should be updated, though it might be better to open a separate issue for that.

Good catch. That was an intentional change, see https://github.com/pandas-dev/pandas/issues/54805 / https://github.com/pandas-dev/pandas/pull/59616, so it is fine to update the docstrings here while updating them to use the string dtype. @arthurlw thanks for the PR! Going to merge this, so we can enable the doctests again. ~Will open an issue for the remaining follow-up task~ (not actually an issue, see inline comment above) Thanks @arthurlw!