PERF: MultiIndex._shallow_copy
- [x] xref #28584, #32568, #32640
- [x] tests added / passed
- [x] passes `black pandas`
- [x] passes `git diff upstream/master -u -- "*.py" | flake8 --diff`
- [x] whatsnew entry

Improves performance of ``MultiIndex._shallow_copy``. Example:

```python
>>> n = 100_000
>>> df = pd.DataFrame({'a': range(n), 'b': range(1, n+1)})
>>> mi = pd.MultiIndex.from_frame(df)
>>> mi.is_lexsorted()
True
>>> mi.get_loc(mi[0])  # also sets up the cache
>>> %timeit mi._shallow_copy().get_loc(mi[0])
8.56 ms ± 127 µs per loop  # master
75.1 µs ± 2.3 µs per loop  # this PR, first commit
46.9 µs ± 792 ns per loop  # this PR, second commit

```

Also adds tests for ``_shallow_copy`` for all index types. This ensures that this issue has been resolved for all index types.
Updated.

It was needlessly complex to do ops in the ``"levels"`` in the cache. Simpler to delete the ``"levels"`` key and just let it be recreated if/when the ``.levels`` attribute is accessed in the new shallow_copied index.
 thanks @topper-123 very nice Yeah, this potentially makes copying thing around much cheaper. Will be interesting to see if there are some specific use cases where performances improve significantly. This PR has introduced a small bug. I'm not sure how deep it goes, but I've encountered it in `PeriodIndex` objects inside a `MultiIndex`. Here is a minimal reproduction:

```python
import pandas as pd
idx = pd.MultiIndex.from_arrays([pd.PeriodIndex([pd.Period("2019Q1"), pd.Period("2019Q2")], name='b')])
idx2 = pd.MultiIndex.from_arrays([idx._get_level_values(level) for level in range(idx.nlevels)])
print(all(x.is_monotonic for x in idx2.levels)) # raises an error
```

There's a simple fix for this, which involves changing `pandas/core/indexes/period.py:260` to `result._cache = {}`. I don't understand the internals enough to know what is causing this issue or whether it extends farther than this case. The problem seems to be that the `_cache` caches the `_engine` attribute, and since we no longer clear the cache when copying the object using `_simple_new`, `_engine` is left in an invalid state. The error is caused by `idx._simple_new()._engine.vgetter()` returning None, which causes issues in `is_monotonic`, among other things. Thanks for the report, @jacobaustin123. This is a bit tricky as you mention, but is caused by the use of ``weakref(self)`` is ``PeriodIndex._engine``, where the weakreffed self in this case is released too early.

This can be solved by deleting ``PeriodIndex._engine`` and falling back to the base ``Index._engine``.

@jbrockmendel , I think you made the new ``Index._engine`` method including ``Index._get_engine_target``. Do you agree that ``PeriodIndex._engine`` should just be deleted?
 > Do you agree that PeriodIndex._engine should just be deleted?

Eventually, yes.  The reason why PeriodIndex._engine exists is because PeriodEngine needs to get the `.freq` attribute of the PeriodIndex.  However, the PeriodEngine methods that actually use `.freq` are never actually reached, because the relevant PeriodIndex methods route around them, and it isn't obvious how intentional that is.

So it should be removed, but there is some untangling to do first. Ok. a fix to the current problem seems to be to replace ``weakref(self)`` with ``weakref(self._values)`` (haven't run the test suite yet, though). We could do that until the fix, I think.