TST (string dtype): add explicit object vs str dtype to index fixture
Also for when we switch the default, that we keep testing object dtype
The two failing builds are failing on main as well, so going to merge this (blocking a follow-up PR) 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 7bd594c81acb5f6428e9ef54ba5a9da1f2860a89
```

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

```
git commit -am 'Backport PR #60116: TST (string dtype): add explicit object vs str dtype to index fixture'
```

4. Push to a named branch:

```
git push YOURFORK 2.3.x:auto-backport-of-pr-60116-on-2.3.x
```

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

> "Backport PR #60116 on branch 2.3.x (TST (string dtype): add explicit object vs str dtype to index fixture)"

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 at https://github.com/pandas-dev/pandas/pull/60136