ERR (string dtype): harmonize setitem error message for python and pyarrow storage

Related to https://github.com/pandas-dev/pandas/issues/60218, but just already making sure we have the same error message for the string dtype for pyarrow vs python storage.
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 692ea6f9d4b05187a05f0811d3241211855d6efb
```

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

```
git commit -am 'Backport PR #60219: ERR (string dtype): harmonize setitem error message for python and pyarrow storage'
```

4. Push to a named branch:

```
git push YOURFORK 2.3.x:auto-backport-of-pr-60219-on-2.3.x
```

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

> "Backport PR #60219 on branch 2.3.x (ERR (string dtype): harmonize setitem error message for python and pyarrow storage)"

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).
                 Manual backport -> https://github.com/pandas-dev/pandas/pull/60232