BUG: PeriodIndex.dtype comparison with string exhibits contradiction
- [x] I have checked that this issue has not already been reported.

- [x] I have confirmed this bug exists on the latest version of pandas.

- [x] (optional) I have confirmed this bug exists on the master branch of pandas.

---

#### Code Sample

```python
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()``

<details>

INSTALLED VERSIONS
------------------
commit           : a34a408e854b6300dbbdd0b29026d19bc5477d73
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
</details>
