Improve MultiIndex label rename checks

[TabLand]

• closes BUG: Pandas column rename function now working for multilevel columns #55169
• Tests added and passed if fixing a bug or adding a new feature
• All code checks passed.
• Added type annotations 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.

[TabLand]

@mroeschke - this is ready for your review

[github-actions]

This pull request is stale because it has been open for thirty days with no activity. Please update and respond to this comment if you're still interested in working on this.

[mroeschke]

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.

[TabLand]

Hi @mroeschke,
The merge conflict was only against the doc/source/whatsnew/v3.0.0.rst file which I've fixed.
After the merge, pandas is failing to compile on my dev environment due to what I'm assuming is an unrelated error:

[9/71] Compiling C object pandas/_libs/interval.cpython-311-x86_64-linux-gnu.so.p/meson-generated_pandas__libs_interval.pyx.c.o ninja: build stopped: subcommand failed.

I'll rebuild my dev environment and update once I can confirm that tests are still passing