TST (string dtype): duplicate pandas/tests/indexes/object tests specifically for string dtypes
We have some `Index` tests specifically for object dtype. Some of those will already get tested through the builds that enable the string dtype by default, but so this PR essentially duplicated those tests and:

- updated the object dtype ones to explicitly use `dtype=object`, and deleted the ones that are specific to strings
- for the duplicated ones that are relevant for strings, parametrized all tests for the various string dtypes (`any_string_dtype` fixture)
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 d8905e4bee2aa0e096ed7831fea7d395d7657120
```

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

```
git commit -am 'Backport PR #60117: TST (string dtype): duplicate pandas/tests/indexes/object tests specifically for string dtypes'
```

4. Push to a named branch:

```
git push YOURFORK 2.3.x:auto-backport-of-pr-60117-on-2.3.x
```

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

> "Backport PR #60117 on branch 2.3.x (TST (string dtype): duplicate pandas/tests/indexes/object tests specifically for string dtypes)"

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 in https://github.com/pandas-dev/pandas/pull/60131