BUG: [ArrowStringArray] Recognize ArrowStringArray in infer_dtype
PR is draft. follow-up to #40679, https://github.com/pandas-dev/pandas/pull/40679#issuecomment-809642594 which is not yet merged and need to make window to run benchmarks https://github.com/pandas-dev/pandas/pull/40679#pullrequestreview-623553887

on master

```
>>> pd.__version__
'1.3.0.dev0+1180.gdd697e1cca'
>>> 
>>> from pandas.core.arrays.string_arrow import ArrowStringDtype
>>> 
>>> arr = ArrowStringDtype.construct_array_type()._from_sequence(["B", pd.NA, "A"])
>>> 
>>> arr
<ArrowStringArray>
['B', <NA>, 'A']
Length: 3, dtype: arrow_string
>>> 
>>> pd._libs.lib.infer_dtype(arr)
'unknown-array'
>>> 
```
> I merged the pre-cursor, so this can be updated now.

Thanks @jorisvandenbossche will get the open PRs updated.

and a couple of PRs for astype failures (both ways) in the pipeline. one uses the fixture https://github.com/pandas-dev/pandas/pull/39908#issuecomment-808905139

and some more parameterization of existing tests where `dtype="string"` https://github.com/pandas-dev/pandas/pull/39908#discussion_r602883047 @simonjayhawkins this can be undrafted? > @simonjayhawkins this can be undrafted?

the benchmarks are running and will have a answer later today. (I don't think this will have any impact on performance, though. And we already did the same change for Period) > (I don't think this will have any impact on performance, though. And we already did the same change for Period)

https://github.com/pandas-dev/pandas/pull/40679#discussion_r603540254 hmm, not good.

```
       before           after         ratio
     [1367cacd]       [37571106]
     <master>         <follow-up>
+        400±50ns         500±50ns     1.25  index_cached_properties.IndexCache.time_inferred_type('Int64Index')
+        400±50ns         500±50ns     1.25  index_cached_properties.IndexCache.time_is_unique('Int64Index')
+     2.69±0.01ms       3.18±0.4ms     1.18  inference.ToDatetimeISO8601.time_iso8601
+      2.20±0.2μs       2.60±0.4μs     1.18  index_cached_properties.IndexCache.time_inferred_type('MultiIndex')
+     1.87±0.04μs      2.11±0.05μs     1.13  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(0, 10000, datetime.timezone.utc)
+     2.00±0.05μs      2.25±0.03μs     1.13  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(0, 12000, None)
+     1.99±0.04μs      2.24±0.01μs     1.13  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(0, 11000, None)
+     2.00±0.02μs      2.26±0.05μs     1.13  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 4000, None)
+     2.00±0.03μs      2.25±0.03μs     1.13  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 1011, None)
+     1.87±0.06μs      2.10±0.01μs     1.12  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(0, 11000, datetime.timezone.utc)
+     2.00±0.03μs      2.24±0.02μs     1.12  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 8000, None)
+     2.01±0.02μs      2.25±0.02μs     1.12  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(0, 6000, None)
+     2.01±0.05μs      2.26±0.02μs     1.12  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 12000, None)
+     2.01±0.05μs      2.25±0.01μs     1.12  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(0, 3000, None)
+     1.99±0.05μs      2.23±0.01μs     1.12  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(0, 1011, None)
+     2.02±0.02μs      2.25±0.01μs     1.12  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(0, 2011, None)
+     2.02±0.02μs      2.25±0.05μs     1.12  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 10000, None)
+      16.3±0.2μs       18.2±0.2μs     1.12  indexing.NumericSeriesIndexing.time_getitem_scalar(<class 'pandas.core.indexes.numeric.Float64Index'>, 'nonunique_monotonic_inc')
+     2.01±0.04μs      2.24±0.02μs     1.12  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 6000, None)
+     2.00±0.02μs      2.23±0.03μs     1.12  tslibs.normalize.Normalize.time_normalize_i8_timestamps(0, tzlocal())
+     2.03±0.03μs      2.27±0.05μs     1.12  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 2011, tzlocal())
+     2.04±0.04μs      2.28±0.08μs     1.11  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 4000, tzlocal())
+     2.00±0.03μs      2.23±0.03μs     1.11  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(0, 8000, None)
+     2.05±0.03μs      2.27±0.05μs     1.11  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(0, 2011, tzlocal())
+     2.01±0.06μs      2.23±0.01μs     1.11  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 7000, None)
+     2.05±0.02μs      2.27±0.03μs     1.11  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(0, 9000, tzlocal())
+     2.01±0.05μs      2.23±0.02μs     1.11  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(0, 2000, None)
+     2.28±0.07μs      2.52±0.01μs     1.11  tslibs.normalize.Normalize.time_normalize_i8_timestamps(100, None)
+     2.04±0.03μs      2.26±0.02μs     1.11  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(0, 1000, None)
+     1.85±0.05μs      2.05±0.03μs     1.11  tslibs.normalize.Normalize.time_normalize_i8_timestamps(1, datetime.timezone.utc)
+     2.01±0.05μs      2.23±0.03μs     1.11  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 11000, None)
+     1.97±0.05μs      2.19±0.03μs     1.11  tslibs.normalize.Normalize.time_normalize_i8_timestamps(1, None)
+     1.89±0.05μs      2.10±0.02μs     1.11  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 6000, datetime.timezone.utc)
+     1.87±0.06μs      2.07±0.01μs     1.11  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(0, 2011, datetime.timezone.utc)
+     2.04±0.03μs      2.25±0.01μs     1.11  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(0, 2000, tzlocal())
+     1.86±0.06μs      2.06±0.02μs     1.11  tslibs.normalize.Normalize.time_normalize_i8_timestamps(0, datetime.timezone.utc)
+     2.07±0.02μs      2.29±0.02μs     1.11  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 1000, tzlocal())
+     2.06±0.04μs      2.27±0.02μs     1.11  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 3000, tzlocal())
+     1.88±0.07μs      2.08±0.01μs     1.11  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 9000, datetime.timezone.utc)
+     1.89±0.05μs      2.09±0.01μs     1.11  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(0, 1011, datetime.timezone.utc)
+     2.02±0.06μs      2.24±0.02μs     1.11  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 4006, None)
+     2.04±0.06μs      2.25±0.02μs     1.11  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 3000, None)
+     1.89±0.06μs      2.09±0.02μs     1.10  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(0, 1000, datetime.timezone.utc)
+     1.89±0.05μs      2.08±0.01μs     1.10  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(0, 7000, datetime.timezone.utc)
+     2.08±0.02μs      2.29±0.01μs     1.10  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 1011, tzlocal())
+     2.05±0.04μs      2.26±0.04μs     1.10  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(0, 11000, tzlocal())
+     1.87±0.04μs      2.06±0.02μs     1.10  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 12000, datetime.timezone.utc)
+     2.05±0.01μs      2.26±0.03μs     1.10  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 9000, tzlocal())
+     2.05±0.03μs      2.26±0.01μs     1.10  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 10000, tzlocal())
+     2.04±0.04μs      2.25±0.03μs     1.10  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 11000, tzlocal())
+     2.03±0.04μs      2.23±0.06μs     1.10  tslibs.normalize.Normalize.time_normalize_i8_timestamps(1, tzlocal())
+     2.04±0.05μs      2.24±0.02μs     1.10  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 6000, tzlocal())
+     2.05±0.03μs      2.25±0.03μs     1.10  tslibs.period.TimeDT64ArrToPeriodArr.time_dt64arr_to_periodarr(1, 5000, tzlocal())
-        55.0±5μs       49.6±0.3μs     0.90  ctors.SeriesConstructors.time_series_constructor(<function no_change at 0x7fdbdd6368b0>, False, 'int')
-      20.4±0.3ms       18.2±0.3ms     0.89  gil.ParallelGroupbyMethods.time_parallel(2, 'mean')
-        64.8±6μs       57.9±0.3μs     0.89  ctors.SeriesConstructors.time_series_constructor(<function no_change at 0x7fdbdd6368b0>, True, 'int')
-       54.3±10μs       46.3±0.8μs     0.85  ctors.SeriesConstructors.time_series_constructor(<function no_change at 0x7fdbdd6368b0>, False, 'float')
-       72.8±20μs       55.6±0.5μs     0.76  ctors.SeriesConstructors.time_series_constructor(<function no_change at 0x7fdbdd6368b0>, True, 'float')

SOME BENCHMARKS HAVE CHANGED SIGNIFICANTLY.
PERFORMANCE DECREASED.
```
will look in detail tomorrow. @simonjayhawkins I think that is mostly within noise-bounds. 
For example, all the `TimeDT64ArrToPeriodArr` are unrelated, AFAIK. The `dt64arr_to_periodarr` function doesn't use infer_dtype, as far as I see, and thus can't be affected by this change. 

This `_TYPE_MAP` is only used for `infer_dtype`. We do have benchmarks for `infer_dtype` (`InferDtype`), but those don't show up in your summary above, so indicating they show no significant change.

We can also check the function itself specifically:

```
In [1]: arr1 = np.array([1, 2, 3])

In [2]: arr2 = pd.array(["a", "b"], dtype="string")

In [3]: %timeit pd.api.types.infer_dtype(arr1)
2.59 µs ± 28.4 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)  <-- master
2.67 µs ± 34.2 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)  <-- PR

In [5]: %timeit pd.api.types.infer_dtype(arr2)
1.04 µs ± 55.3 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)  <-- master
1.4 µs ± 41.8 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)  <-- PR
```

So for a StringArray input, there is some change. But even then, the question is whether this is change is significant for actual usage. We typically use `infer_dtype` when handling generic input (eg in constructors), and for those cases this change will not be significant, I think.  thanks @simonjayhawkins 