Use NaN as na_value for new pyarrow_numpy StringDtype
- [ ] closes #xxxx (Replace xxxx with the GitHub issue number)
- [ ] [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
- [ ] 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.

xref https://github.com/pandas-dev/pandas/issues/54792
Looking just at the single commit https://github.com/pandas-dev/pandas/pull/54585/commits/d862ecaf3aa05082d36512f994d99106cc7c10b1, that looks good to me cc @jbrockmendel  rebased, the diff is more manageable now

cc @jorisvandenbossche @jbrockmendel  merging this @meeseeksdev backport to 2.1.x