BUG: PeriodIndex.dtype comparison with string exhibits contradiction

[hickmanw]

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

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

• (optional) I have confirmed this bug exists on the master branch of pandas.

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Code Sample

index = pd.period_range(start='01Jan2019', end='01Dec2019', freq='M')
string = "period[M]"
index.dtype == string  # True
index.dtype != string  # also True

Problem description

PeriodIndex.dtype is both __eq__ and __ne__ to the string representation of its frequency. I suspect that comparing the dtype to a string is not recommended, but the contradiction seemed worth reporting. I don't know when this was introduced, but it did not occur in 0.25.3.

Expected Output

One of the two comparisons to evaluate to False.

Output of pd.show_versions()

INSTALLED VERSIONS

commit : a34a408
python : 3.8.6.final.0
python-bits : 64
OS : Linux
OS-release : 4.14.198-152.320.amzn2.x86_64
Version : #1 SMP Wed Sep 23 23:57:28 UTC 2020
machine : x86_64
processor : x86_64
byteorder : little
LC_ALL : en_US.UTF-8
LANG : en_US.UTF-8
LOCALE : en_US.UTF-8

pandas : 1.2.0.dev0+830.ga34a408e8
numpy : 1.19.2
pytz : 2020.1
dateutil : 2.8.1
pip : 20.2.4
setuptools : 49.6.0.post20201009
Cython : 0.29.21
pytest : 6.1.1
hypothesis : 5.37.3
sphinx : 3.2.1
blosc : None
feather : None
xlsxwriter : 1.3.7
lxml.etree : 4.6.1
html5lib : 1.1
pymysql : None
psycopg2 : None
jinja2 : 2.11.2
IPython : 7.18.1
pandas_datareader: None
bs4 : 4.9.3
bottleneck : 1.3.2
fsspec : 0.8.4
fastparquet : 0.4.1
gcsfs : 0.7.1
matplotlib : 3.3.2
numexpr : 2.7.1
odfpy : None
openpyxl : 3.0.5
pandas_gbq : None
pyarrow : 1.0.1
pyxlsb : None
s3fs : 0.4.2
scipy : 1.5.2
sqlalchemy : 1.3.20
tables : 3.6.1
tabulate : 0.8.7
xarray : 0.16.1
xlrd : 1.2.0
xlwt : 1.3.0
numba : 0.51.2

[dsaxton]

Thanks, this is a regression due to #34590

cc @jbrockmendel