BUG: PeriodDtype incorrectly caching different DateOffset freqs
- [x] closes #24121 (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
- [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/vX.X.X.rst` file if fixing a bug or adding a new feature.

pandas/core/dtypes/dtypes.py:902: error: Incompatible types in assignment (expression has type "Dict[int, PandasExtensionDtype]", base class "PandasExtensionDtype" defined the type as "Dict[str, PandasExtensionDtype]")  [assignment]
Found 1 error in 1 file (checked 1417 source files)

This is one of the complaints I've been seeing locally for a while.  No idea why its showing up now thanks @mroeschke 