ENH: EA._hash_pandas_object
- [x] closes #51108 (Replace xxxx with the GitHub issue number)
- [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
- [ ] 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.
- [ ] Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature.

@MarcoGorelli any thoughts here? I'll take a look this week, thanks for the ping @phofl thoughts? Also, does it need adding here

https://github.com/pandas-dev/pandas/blob/089aa0cae6e66814a98749aba6927c9267182457/pandas/core/arrays/base.py#L117-L146  

? > Also, does it need adding here

Yes, will update the docs job is failing , showing

```
2023-02-27T22:46:43.2184483Z /home/runner/work/pandas/pandas/pandas/core/arrays/base.py:docstring of pandas.core.arrays.base.ExtensionArray:116: WARNING: autosummary: stub file not found 'pandas.api.extensions.ExtensionArray._hash_pandas_object'. Check your autosummary_generate setting.
```

seems related? its complaining about No Examples section found?  how do i tell it to ignore that? the ignore list is in code_checks.sh thx @jbrockmendel 