BUG: Divide zero by zero returns NaN in a Float64 dtype column which cannot be handled by fillna

[jaspersival]

• 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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Note: Please read this guide detailing how to provide the necessary information for us to reproduce your bug.

Code Sample, a copy-pastable example

series1 = pd.Series([0.], dtype=pd.Float64Dtype())
series2 = pd.Series([0.], dtype=pd.Float64Dtype())
series1/series2
0    NaN
dtype: Float64
(series1/series2).fillna(0)
0    NaN
dtype: Float64

Problem description

When dividing a pd.Float64Dtype series by another of the same type, and if there is a zero by zero division taking place, then pandas returns a NaN value for that particular case. If I want to replace this NaN value with something else, I cannot do that.
I can only get it to work if I do:

(series1/series2).astype(float).astype(pd.Float64Dtype()).fillna(0)
0    0.0
dtype: Float64

Expected Output

(series1/series2).fillna(0)
0    0.0
dtype: Float64

Output of pd.show_versions()

INSTALLED VERSIONS

commit : f2c8480
python : 3.9.1.final.0
python-bits : 64
OS : Windows
OS-release : 10
Version : 10.0.19041
machine : AMD64
processor : Intel64 Family 6 Model 85 Stepping 4, GenuineIntel
byteorder : little
LC_ALL : None
LANG : None
LOCALE : English_United States.1252
pandas : 1.2.3
numpy : 1.20.2
pytz : 2021.1
dateutil : 2.8.1
pip : 21.0.1
setuptools : 52.0.0
Cython : None
pytest : 6.2.2
hypothesis : None
sphinx : None
blosc : None
feather : None
xlsxwriter : None
lxml.etree : None
html5lib : None
pymysql : None
psycopg2 : None
jinja2 : 2.11.3
IPython : None
pandas_datareader: None
bs4 : None
bottleneck : None
fsspec : None
fastparquet : None
gcsfs : None
matplotlib : None
numexpr : None
odfpy : None
openpyxl : 3.0.7
pandas_gbq : None
pyarrow : None
pyxlsb : None
s3fs : None
scipy : 1.6.1
sqlalchemy : 1.4.3
tables : None
tabulate : None
xarray : None
xlrd : 1.2.0
xlwt : None
numba : None

[attack68]

lots of discussion on this: #32265 and #30188

[mzeitlin11]

And #39926, will close in favor of discussion there and other issues mentioned