BUG: Fix Index.equals between object and string
- [X] closes #61099 
- [X] [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
- [X] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [ ] Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) to new arguments/methods/functions.
- [X] Added an entry in the latest `doc/source/whatsnew/v3.0.0.rst` file if fixing a bug or adding a new feature.

## Description of the code change on `Index.equals`

On the main branch, `Index.equals` casts `self` to `object` only when `self.dtype.na_value` is `np.nan`. The comparison actually succeeds when `self.dtype.na_value` is `np.nan` as below.
```python
>>> import pandas as pd
>>> import numpy as np

>>> s1 = pd.Series([1, 2, 3], index=['a', 'b', 'c'])
>>> s2 = pd.Series([4, 5, 6], index=['a', 'b', 'c'])
>>> s2.index = s2.index.astype(pd.StringDtype(storage="pyarrow", na_value=np.nan))

>>> print(s1 < s2)
a    True
b    True
c    True
dtype: bool
```
However, since doc stated that `dtype` is not compared, `self` should be casted regardless of `self.dtype.na_value` so that `self` could be compared with other dtypes as desired.


## Description of the code change on `test_mixed_col_index_dtype`

`using_infer_string` has been removed since I think that `result` should be `string` regardless of `using_infer_string`. This is becaus of the code change made on `Index.equals` - since `Index.equals` consider `df1.columns` is equal to `df2.colums`, `Index.intersection` returns `self`(which is `string`). You could see the result becomes `object`(which is the dtype of `df2`) in case of `result = df2 + df1`. On the main branch, on the other hand, `Index.intersection` returns `object` because `Index.equals` returns `False`, and then both `self` and `other` are cast to `object` by `_find_common_type_compat`. (see `L3287` at pandas/core/indexes/base.py)

https://github.com/pandas-dev/pandas/blob/25e64629ec317fba2fc1c2834b20362fa6c1fd89/pandas/core/indexes/base.py#L3286-L3290


* I created this pull request since @MayurKishorKumar doesn't seem to work on this issue anymore, but please let me know if there is going to be further actions on the previous PR and I am supposed to close this one.
Thanks @sanggon6107!