API / CoW: shallow copy of DataFrame/Series (.copy(deep=False)) also returns shallow copy of the index/columns
Broken off a subset of https://github.com/pandas-dev/pandas/pull/53699, only focusing on the behaviour of `df.copy(deep=False)` (i.e. a "shallow copy"). And address the `copy(deep=False)` part of https://github.com/pandas-dev/pandas/issues/53721.

Currently, a shallow copy of a DataFrame/Series only creates a new DataFrame/Series object, but still shares the same data (with tracking references, to ensure copy-on-write) and Index objects. 
Because Index objects have some mutable aspects (not their values, but for example there `name`), that means that sharing an identical object in the resulting shallow copy can propagate mutations to those mutable parts of the index.

See https://github.com/pandas-dev/pandas/issues/53721 for a longer description of the context.

This PR just updates the DataFrame/Series `copy()` method to also take a shallow copy of the index/columns Index object, in case of `deep=False`.

And based on the tests I had to update to get this passing, it illustrates some of the consequential changes in other places because of changing the shallow copy.


- [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).
- [x] 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.

Thanks could use a whatsnew note in 2.1 This pull request is stale because it has been open for thirty days with no activity. Please [update](https://pandas.pydata.org/pandas-docs/stable/development/contributing.html#updating-your-pull-request) and respond to this comment if you're still interested in working on this. Thanks @jorisvandenbossche 