BUG: Fix getitem dtype preservation with multiindexes
- [x] closes #51261
- [x] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests)
- [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.~~ NA
- [ ] Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature. Will wait and see if approach is okay first before changelog.

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 for the pull request, but it appears to have gone stale. If interested in continuing, please merge in the main branch, address any review comments and/or failing tests, and we can reopen. This is actually waiting on an approval / merge (or more comments), although needs a merge of main now > Could you add a whatsnew entry in `v.2.1.0.rst`?

Sure, I've had a go at this, I originally left it because I wasn't sure how to clearly decribe what's actually being fixed. Thanks @m-richards 