BUG: Pandas `string` dtype reverts to `object` dtype when initialising index
- [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.

1.
```
In [2]: pd.Index(['A', 'B', 'C', 'D'], dtype=pd.StringDtype()).dtype
Out[2]: dtype('O')

In [3]: pd.Index([1, 2, 3, 4], dtype=int).dtype
Out[3]: dtype('int64')
```
2.
<details>
  <summary>Output pd.show_versions()</summary>


```
  INSTALLED VERSIONS
  ------------------
  commit           : f00ed8f47020034e752baf0250483053340971b0
  python           : 3.7.9.final.0
  python-bits      : 64
  OS               : Windows
  OS-release       : 10
  Version          : 10.0.19041
  machine          : AMD64
  processor        : Intel64 Family 6 Model 58 Stepping 0, GenuineIntel
  byteorder        : little
  LC_ALL           : None
  LANG             : None
  LOCALE           : None.None
  
  pandas           : 1.3.0
  numpy            : 1.19.5
  pytz             : 2020.5
  dateutil         : 2.8.1
  pip              : 21.0
  setuptools       : 49.6.0.post20210108
  Cython           : None
  pytest           : 6.2.1
  hypothesis       : None
  sphinx           : 3.5.4
  blosc            : None
  feather          : None
  xlsxwriter       : 1.3.9
  lxml.etree       : 4.6.3
  html5lib         : None
  pymysql          : None
  psycopg2         : 2.8.6 (dt dec pq3 ext lo64)
  jinja2           : 2.11.2
  IPython          : 7.19.0
  pandas_datareader: None
  bs4              : 4.9.3
  bottleneck       : None
  fsspec           : 2021.05.0
  fastparquet      : None
  gcsfs            : None
  matplotlib       : 3.3.3
  numexpr          : None
  odfpy            : None
  openpyxl         : 3.0.7
  pandas_gbq       : None
  pyarrow          : 3.0.0
  pyxlsb           : None
  s3fs             : None
  scipy            : 1.6.2
  sqlalchemy       : 1.4.3
  tables           : None
  tabulate         : 0.8.9
  xarray           : 0.16.2
  xlrd             : None
  xlwt             : None
  numba            : 0.53.1
```

</details>


3.
I don't think the dtype should change here. Seems to be okay for other dtypes apart from strings.