BUG: Confusing conversion from string?

[kwhkim]

Pandas version checks

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

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

• I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

import pandas as pd

pd.Timestamp('-111-01-01') == pd.Timestamp('111-01-01')
# True

pd.Timestamp('-0111-01-01') == pd.Timestamp('0111-01-01')
# False

np.datetime64('-111-01-01') == np.datetime64('111-01-01')
# False

np.datetime64('-0111-01-01') == np.datetime64('0111-01-01')
# False

Issue Description

For the first example, it should raise "ValueError: time data does not match format '%Y-%m-%d'" or should be False

Expected Behavior

False

Installed Versions

Details

INSTALLED VERSIONS

commit : 2a953cf
python : 3.12.0.final.0
python-bits : 64
OS : Darwin
OS-release : 20.6.0
Version : Darwin Kernel Version 20.6.0: Mon Aug 29 04:31:12 PDT 2022; root:xnu-7195.141.39~2/RELEASE_ARM64_T8101
machine : arm64
processor : arm
byteorder : little
LC_ALL : None
LANG : ko_KR.UTF-8
LOCALE : ko_KR.UTF-8

pandas : 2.1.3
numpy : 1.26.0
pytz : 2023.3.post1
dateutil : 2.8.2
setuptools : 68.2.2
pip : 23.3.1
Cython : None
pytest : None
hypothesis : None
sphinx : None
blosc : None
feather : None
xlsxwriter : None
lxml.etree : None
html5lib : None
pymysql : None
psycopg2 : None
jinja2 : 3.1.2
IPython : 8.17.2
pandas_datareader : None
bs4 : 4.12.2
bottleneck : None
dataframe-api-compat: None
fastparquet : None
fsspec : None
gcsfs : None
matplotlib : None
numba : None
numexpr : None
odfpy : None
openpyxl : None
pandas_gbq : None
pyarrow : None
pyreadstat : None
pyxlsb : None
s3fs : None
scipy : None
sqlalchemy : None
tables : None
tabulate : None
xarray : None
xlrd : None
zstandard : None
tzdata : 2023.3
qtpy : None
pyqt5 : None

[rhshadrach]

Confirmed on main, further investigations and PRs to fix are welcome!