BUG: __eq__ raises NotImplementedError for new arrow string dtype

[phofl]

Reproducible Example

ser = pd.Series([False, True])
ser2 = pd.Series(["a", "b"], dtype="string[pyarrow_numpy]")
ser == ser2

Issue Description

This shouldn't raise if we want to emulate object dtype behavior

Expected Behavior

all False return value
cc @jorisvandenbossche thoughts?

Installed Versions

Replace this line with the output of pd.show_versions()

[jorisvandenbossche]

Regardless of exactly emulating object dtype, we should maybe ask ourselves the more general question: how does pandas handle equality comparisons of incompatible dtypes in general? (or at least with the non-arrow dtypes)

At the moment, it seems we just never raise an error, with one exception for comparing two categoricals (based on a quick experiment, didn't include every possible dtype):

serieses =[
    pd.Series(["a", "b"]),
    pd.Series(["a", "b"], dtype="string"),
    pd.Series([True, False]),
    pd.Series([True, False], dtype="boolean"),
    pd.Series([1, 2]),
    pd.Series([1, 2], dtype="Int64"),
    pd.Series([0.1, 0.2]),
    pd.Series([0.1, 0.2], dtype="Float64"),
    pd.Series(pd.date_range("2012-01-01", periods=2)),
    pd.Series(pd.timedelta_range("1 days", periods=2)),
    pd.Series(pd.period_range("2012", periods=2)),
    pd.Series(['a', 'b'], dtype="category"),
    pd.Series([1, 2], dtype="category"),
]

for left in serieses:
    for right in serieses:
        try:
            left == right
        except:
            print(f"Exception for {left.dtype} and {right.dtype}")

gives

Exception for category and category
Exception for category and category

So if we want to keep that "rule" consistent, then I think the new default string dtype should also never raise in comparisons, but give Falses instead.

[jbrockmendel]

in general we follow python semantics for non-comparable dtypes: == returns all-False, != returns all-True, and inequalities raise. A boilerplate version of this logic is in ops.invalid_comparison