TST(string dtype): Resolve some HDF5 xfails
- [ ] 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).
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The behavior here is that if you write out strings using object dtype, then you still get the inferred string dtype when reading back in. That is, we (likely) do not round-trip dtypes if the value of `future.infer_string` is different when reading/writing.
Owee, I'm MrMeeseeks, Look at me.

There seem to be a conflict, please backport manually. Here are approximate instructions:

1. Checkout backport branch and update it.

```
git checkout 2.3.x
git pull
```

2. Cherry pick the first parent branch of the this PR on top of the older branch:
```
git cherry-pick -x -m1 2edc7c9ad9a8b2e1f8df981def5b5b0c434d9ab0
```

3. You will likely have some merge/cherry-pick conflict here, fix them and commit:

```
git commit -am 'Backport PR #60615: TST(string dtype): Resolve some HDF5 xfails'
```

4. Push to a named branch:

```
git push YOURFORK 2.3.x:auto-backport-of-pr-60615-on-2.3.x
```

5. Create a PR against branch 2.3.x, I would have named this PR:

> "Backport PR #60615 on branch 2.3.x (TST(string dtype): Resolve some HDF5 xfails)"

And apply the correct labels and milestones.

Congratulations — you did some good work! Hopefully your backport PR will be tested by the continuous integration and merged soon!

Remember to remove the `Still Needs Manual Backport` label once the PR gets merged.

If these instructions are inaccurate, feel free to [suggest an improvement](https://github.com/MeeseeksBox/MeeseeksDev).
                 Thanks @rhshadrach  Opened manual backport: #60626