BUG: transpose inferring dtype for dt in object column
- [x] closes #51546 (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).
- [x] 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.

cc @jbrockmendel thoughts about doing it for 2.0 or 2.1?
> thoughts about doing it for 2.0 or 2.1?

no strong preference.  i think we're doing more still-merging-to-2.0 in the RC this time around than we usually do, not sure how much is too much Then lets do for 2.1, no strong preference either on my side thanks @phofl 