BUG: Series.str.split broken with pyarrow strings and regex argument

[WillAyd]

Pandas version checks

• I have checked that this issue has not already been reported.

• I have confirmed this bug exists on the latest version of pandas.

• I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

In [70]: pd.Series(
    ...:     ["230/270/270", "240-290-290"],
    ...:     dtype="string[pyarrow]"
    ...: ).str.split(r"/|-", expand=True)
Out[70]: 
     0    1    2
0  230  270  270
1  240  290  290

In [71]: pd.Series(
    ...:     ["230/270/270", "240-290-290"],
    ...:     dtype=pd.ArrowDtype(pa.string())
    ...: ).str.split(r"/|-", expand=True)
Out[71]: 
             0
0  230/270/270
1  240-290-290

Issue Description

It doesn't look like arrow strings work when using a regular expression argument to split. I am also a bit confused why there is a difference between string[pyarrow] and pd.ArrowDtype(pa.string())

@phofl in case you know what's going on

Expected Behavior

Values should split for arrow string type

Installed Versions

In [73]: pd.version
Out[73]: '3.0.0.dev0+681.g434fda08cf'

[asishm]

iirc
pd.ArrowDtype(pa.string()) goes through ./pandas/core/arrays/arrow/array.py
while string[pyarrow] goes through ./pandas/core/arrays/string_arrow.py

[yuanx749]

As per https://pandas.pydata.org/docs/user_guide/pyarrow.html

The string alias "string[pyarrow]" maps to pd.StringDtype("pyarrow") which is not equivalent to specifying dtype=pd.ArrowDtype(pa.string())

[yuanx749]

take