DOC: `.to_string` `float_format` behavior inconsistent with documentation and different for extension types

Pandas version checks

• I have checked that the issue still exists on the latest versions of the docs on main here

Location of the documentation

https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.to_string.html

Documentation problem

DataFrame.to_string docstring says float_format should be "one-parameter function, optional, default None", but passing a format string (e.g. '%.3e') works too, like in .to_csv, for NumPy float type columns/series.

However, unlike .to_csv, .to_string with float_format string fails with TypeError: 'str' object is not callable for float extension type columns/series.

s = pd.Series([1.1, 2.2, 3.3])

# works
s.to_csv(float_format="%.3e")
s.convert_dtypes().to_csv(float_format="%.3e")
s.to_string(float_format="%.3e")

# fails
s.convert_dtypes().to_string(float_format="%.3e")

xref: #9448 https://github.com/pandas-dev/pandas-stubs/issues/730

Suggested fix for documentation

It would be nice for the documentation to reflect that a format string can be used, but probably the issue with extension types should be fixed first.

Note also that FloatFormatType, used in the internal typing of DataFrame.to_string,

https://github.com/pandas-dev/pandas/blob/abcd440e11d061171b0a4eeb058748629a288568/pandas/core/frame.py#L1149

does include str as an option.

https://github.com/pandas-dev/pandas/blob/abcd440e11d061171b0a4eeb058748629a288568/pandas/_typing.py#L303