BUG: Some string methods treat "." as regex, others don't
- [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.

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

---

#### Code Sample, a copy-pastable example

```python
import re
import pandas as pd

texts = ['aa.aa', 'bbb']

# str.replace vs re.sub: '.' is not treated as regex by pandas
print(*pd.Series(texts).str.replace('.', '*'), sep=', ')
print(*pd.Series(texts).str.replace('..', '**'), sep=', ')
print(*[re.sub('.', '*', t) for t in texts], sep=', ')
print(*[re.sub('..', '**', t) for t in texts], sep=', ')
print()

# str.split vs re.split: '.' is not treated as regex by pandas
print(*pd.Series(texts).str.split('.'), sep=', ')
print(*pd.Series(texts).str.split('..'), sep=', ')
print(*[re.split('.', t) for t in texts], sep=', ')
print(*[re.split('..', t) for t in texts], sep=', ')
print()

# str.contains vs re.search: '.' is treated as regex by both
print(*pd.Series(texts).str.contains('.'), sep=', ')
print(*pd.Series(texts).str.contains('..'), sep=', ')
print(*[re.search('.', t).start() for t in texts], sep=', ')
print(*[re.search('..', t).start() for t in texts], sep=', ')
```

#### Problem description

Currently some pandas string functions (eg str.replace, str.split) treat `"."` as a regex matching any character, while some others (eg str.contains) treat it like `"[.]"`, the regex matching the "dot" character. The `".."` string is instead always correctly handled as the regex matching any two characters.

#### Expected Output

pandas string functions (when in regex mode) should treat `"."` consistently as the regex matching any single character. If there is a reason to treat as an exception, or to do so only for some string functions, should be documented.

#### Output of ``pd.show_versions()``

<details>

INSTALLED VERSIONS
------------------
commit           : db08276bc116c438d3fdee492026f8223584c477
python           : 3.8.5.final.0
python-bits      : 64
OS               : Darwin
OS-release       : 19.6.0
Version          : Darwin Kernel Version 19.6.0: Thu Jun 18 20:49:00 PDT 2020; root:xnu-6153.141.1~1/RELEASE_X86_64
machine          : x86_64
processor        : i386
byteorder        : little
LC_ALL           : None
LANG             : None
LOCALE           : None.UTF-8

pandas           : 1.1.3
numpy            : 1.19.2
pytz             : 2020.1
dateutil         : 2.8.1
pip              : 20.2.4
setuptools       : 50.3.1.post20201107
Cython           : None
pytest           : None
hypothesis       : None
sphinx           : None
blosc            : None
feather          : None
xlsxwriter       : None
lxml.etree       : None
html5lib         : None
pymysql          : None
psycopg2         : None
jinja2           : None
IPython          : 7.18.1
pandas_datareader: None
bs4              : None
bottleneck       : None
fsspec           : None
fastparquet      : None
gcsfs            : None
matplotlib       : None
numexpr          : None
odfpy            : None
openpyxl         : None
pandas_gbq       : None
pyarrow          : None
pytables         : None
pyxlsb           : None
s3fs             : None
scipy            : None
sqlalchemy       : None
tables           : None
tabulate         : None
xarray           : None
xlrd             : None
xlwt             : None
numba            : None

</details>
