Output formatting: preserve quoting for string categories
POC for https://github.com/pandas-dev/pandas/issues/61890 if we want to keep the quoting behaviour
It's an easy change, so let's add to 2.3.x (I might just not backport all test fixes) 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 c849d39c4c956495c0c86b85ab561bf74bf5df8d
```

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

```
git commit -am 'Backport PR #61891: Output formatting: preserve quoting for string categories'
```

4. Push to a named branch:

```
git push YOURFORK 2.3.x:auto-backport-of-pr-61891-on-2.3.x
```

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

> "Backport PR #61891 on branch 2.3.x (Output formatting: preserve quoting for string categories)"

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/61966