TST (string dtype): add test build with future strings enabled without pyarrow
~On top of the commits of https://github.com/pandas-dev/pandas/pull/58451 and https://github.com/pandas-dev/pandas/pull/59388 for now.~  

Follow-up on https://github.com/pandas-dev/pandas/pull/58451 which implemented the `StringDtype("python", na_value=np.nan)` variant, this PR is adding a custom CI test build for it (future option enabled but without pyarrow installed, such that it uses that new dtype by default)

xref https://github.com/pandas-dev/pandas/issues/54792
Thanks @jorisvandenbossche 