is_datetimelike treatment of PeriodDtype inconsistent

[jbrockmendel]

xref #22137

def is_datetimelike(arr):
    """ [...] """    
    return (is_datetime64_dtype(arr) or is_datetime64tz_dtype(arr) or
            is_timedelta64_dtype(arr) or
            isinstance(arr, ABCPeriodIndex) or
            is_datetimetz(arr))

is_datetimetz is AFAICT redundant with is_datetime64tz_dtype, so ignore that part for now. Shouldn't the ABCPeriodIndex check be the more general is_period_dtype? Or for that matter, is_period_arraylike?

cc: @TomAugspurger

[TomAugspurger]

Yeah, we shouldn't be special casing PeriodIndex there.