Init arguments upon multiple inheritance from LightningDataModule and other parent class

Nobody has claimed this yet.

### Bug description

When defining a new class inheriting from both `lightning.pytorch.LightningDataModule` and a user-defined parent class, a TypeError is raised upon initialization of an instance of the new class.

Expected behavior: no TypeError when inheriting from `lightning.pytorch.LightningDataModule` (see Check2 in the MRE) and [docs](https://lightning.ai/docs/pytorch/stable/data/datamodule.html).

### What version are you seeing the problem on?

2.0+

### How to reproduce the bug

```python
from lightning.pytorch import LightningDataModule

class Base:
def __init__(self, foo):
print(foo)

class Other:
def __init__(self):
pass

class Check1(Other, Base):
def __init__(self, config):
Other.__init__(self)
Base.__init__(self, **config)

class Check2(LightningDataModule):
def __init__(self):
LightningDataModule.__init__(self)

class CustomDataModule(LightningDataModule, Base):
def __init__(self, config):
LightningDataModule.__init__(self)
Base.__init__(self, **config)

if __name__ == "__main__":
config = {"foo": "foo"}
Check1(config)
Check2()
CustomDataModule(config)
```

### Error messages and logs

```
Traceback (most recent call last):
File ".../debug.py", line 35, in
CustomDataModule(config)
File ".../debug.py", line 27, in __init__
LightningDataModule.__init__(self)
File ".../.pyenv/versions/3.8.10/envs/frm_test/lib/python3.8/site-packages/lightning/pytorch/core/datamodule.py", line 64, in __init__
super().__init__()
File .../.pyenv/versions/3.8.10/envs/frm_test/lib/python3.8/site-packages/lightning/pytorch/core/hooks.py", line 298, in __init__
super().__init__()
File ".../.pyenv/versions/3.8.10/envs/frm_test/lib/python3.8/site-packages/lightning/pytorch/core/mixins/hparams_mixin.py", line 31, in __init__
super().__init__()
TypeError: __init__() missing 1 required positional argument: 'foo'
```

### Environment

Current environment

```
* CUDA:
- GPU: None
- available: False
- version: 11.7
* Lightning:
- lightning: 2.0.1.post0
- lightning-cloud: 0.5.32
- lightning-utilities: 0.8.0
- pytorch-lightning: 2.0.0
- torch: 2.0.0
- torch-tb-profiler: 0.4.1
- torchmetrics: 0.11.4
- torchsummary: 1.5.1
- torchview: 0.2.6
- torchvision: 0.15.1
* Packages:
- absl-py: 1.4.0
- affine: 2.4.0
- aiofiles: 22.1.0
- aiohttp: 3.8.4
- aiosignal: 1.3.1
- aiosqlite: 0.18.0
- alembic: 1.9.2
- anyio: 3.6.2
- argon2-cffi: 21.3.0
- argon2-cffi-bindings: 21.2.0
- arrow: 1.2.3
- asciitree: 0.3.3
- asttokens: 2.0.8
- astunparse: 1.6.3
- async-timeout: 4.0.2
- attrs: 22.1.0
- autopage: 0.5.1
- babel: 2.10.3
- backcall: 0.2.0
- backoff: 2.2.1
- beautifulsoup4: 4.11.1
- bleach: 5.0.1
- blessed: 1.20.0
- bokeh: 2.4.3
- bottle: 0.12.25
- bqplot: 0.12.36
- branca: 0.5.0
- cachetools: 5.2.0
- certifi: 2022.9.24
- cffi: 1.15.1
- cftime: 1.6.2
- charset-normalizer: 2.1.1
- click: 8.1.3
- click-plugins: 1.1.1
- cliff: 4.2.0
- cligj: 0.7.2
- cloudpickle: 2.2.1
- cmaes: 0.9.1
- cmake: 3.26.0
- cmd2: 2.4.3
- colorlog: 6.7.0
- colour: 0.1.5
- comm: 0.1.2
- compress-json: 1.0.8
- contourpy: 1.0.5
- croniter: 1.3.8
- cycler: 0.11.0
- dask: 2022.12.0
- dask-optuna: 0.0.2
- dateutils: 0.6.12
- debugpy: 1.6.3
- decorator: 5.1.1
- deepdiff: 6.3.0
- defusedxml: 0.7.1
- dill: 0.3.6
- distributed: 2022.12.0
- dnspython: 2.3.0
- docopt: 0.6.2
- earthengine-api: 0.1.343
- ee-extra: 0.0.14
- eerepr: 0.0.4
- email-validator: 1.3.1
- entrypoints: 0.4
- executing: 1.1.1
- fastapi: 0.88.0
- fasteners: 0.18
- fastjsonschema: 2.16.2
- ffmpeg-python: 0.2.0
- filelock: 3.8.0
- fiona: 1.8.22
- flask: 2.2.2
- flatbuffers: 23.1.21
- folium: 0.13.0
- fonttools: 4.38.0
- fqdn: 1.5.1
- frarssmap: 0.0.0
- frozenlist: 1.3.3
- fsspec: 2023.1.0
- future: 0.18.2
- gast: 0.4.0
- gdown: 4.5.3
- geeadd: 0.5.6
- geemap: 0.17.1
- geocoder: 1.38.1
- geojson: 2.5.0
- geopandas: 0.10.2
- google-api-core: 2.10.2
- google-api-python-client: 1.12.11
- google-auth: 2.17.2
- google-auth-httplib2: 0.1.0
- google-auth-oauthlib: 0.4.6
- google-cloud-core: 2.3.2
- google-cloud-storage: 2.5.0
- google-crc32c: 1.5.0
- google-pasta: 0.2.0
- google-resumable-media: 2.4.0
- googleapis-common-protos: 1.56.4
- greenlet: 2.0.2
- grip: 4.6.1
- grpcio: 1.51.1
- h11: 0.14.0
- h3: 3.7.4
- h3pandas: 0.2.3
- h5py: 3.8.0
- heapdict: 1.0.1
- httpcore: 0.16.3
- httplib2: 0.20.4
- httplib2shim: 0.0.3
- httptools: 0.5.0
- httpx: 0.23.3
- idna: 3.4
- importlib-metadata: 5.0.0
- importlib-resources: 5.10.0
- inquirer: 3.1.3
- ipyevents: 2.0.1
- ipyfilechooser: 0.6.0
- ipykernel: 6.16.2
- ipyleaflet: 0.17.2
- ipython: 8.5.0
- ipython-genutils: 0.2.0
- ipytree: 0.2.2
- ipywidgets: 7.6.5
- isoduration: 20.11.0
- itsdangerous: 2.1.2
- jedi: 0.18.1
- jinja2: 3.1.2
- joblib: 1.2.0
- json5: 0.9.10
- jsonpointer: 2.3
- jsonschema: 4.16.0
- jupyter: 1.0.0
- jupyter-client: 7.1.0
- jupyter-console: 6.4.4
- jupyter-contrib-core: 0.4.2
- jupyter-contrib-nbextensions: 0.5.1
- jupyter-core: 4.9.1
- jupyter-events: 0.5.0
- jupyter-highlight-selected-word: 0.2.0
- jupyter-latex-envs: 1.4.6
- jupyter-nbextensions-configurator: 0.6.1
- jupyter-server: 1.13.1
- jupyter-server-fileid: 0.6.0
- jupyter-server-terminals: 0.4.4
- jupyter-server-ydoc: 0.6.1
- jupyter-ydoc: 0.2.2
- jupyterlab: 3.2.5
- jupyterlab-pygments: 0.1.2
- jupyterlab-server: 2.10.2
- jupyterlab-sublime: 0.4.1
- jupyterlab-widgets: 1.0.2
- kaleido: 0.2.1
- keras: 2.11.0
- kiwisolver: 1.4.4
- libclang: 15.0.6.1
- lightgbm: 3.3.5
- lightning: 2.0.1.post0
- lightning-cloud: 0.5.32
- lightning-utilities: 0.8.0
- lit: 16.0.0
- locket: 1.0.0
- logzero: 1.7.0
- lxml: 4.9.2
- mako: 1.2.4
- markdown: 3.4.1
- markdown-it-py: 2.2.0
- markupsafe: 2.1.1
- matplotlib: 3.6.3
- matplotlib-inline: 0.1.6
- mdurl: 0.1.2
- mistune: 0.8.4
- mpmath: 1.3.0
- msgpack: 1.0.4
- multidict: 6.0.4
- multiprocess: 0.70.14
- munch: 2.5.0
- nbclassic: 0.3.4
- nbclient: 0.5.9
- nbconvert: 6.4.0
- nbformat: 5.1.3
- nest-asyncio: 1.5.6
- netcdf4: 1.6.2
- netron: 6.5.5
- networkx: 3.0
- notebook: 6.1.5
- notebook-shim: 0.2.0
- numcodecs: 0.11.0
- numpy: 1.23.4
- nvidia-cublas-cu11: 11.10.3.66
- nvidia-cuda-cupti-cu11: 11.7.101
- nvidia-cuda-nvrtc-cu11: 11.7.99
- nvidia-cuda-runtime-cu11: 11.7.99
- nvidia-cudnn-cu11: 8.5.0.96
- nvidia-cufft-cu11: 10.9.0.58
- nvidia-curand-cu11: 10.2.10.91
- nvidia-cusolver-cu11: 11.4.0.1
- nvidia-cusparse-cu11: 11.7.4.91
- nvidia-nccl-cu11: 2.14.3
- nvidia-nvtx-cu11: 11.7.91
- oauthlib: 3.2.2
- opt-einsum: 3.3.0
- optuna: 3.1.0
- optuna-dashboard: 0.9.0
- ordered-set: 4.1.0
- orjson: 3.8.8
- packaging: 21.3
- pandas: 1.5.1
- pandocfilters: 1.5.0
- parso: 0.8.3
- partd: 1.3.0
- path-and-address: 2.0.1
- pathos: 0.3.0
- pbr: 5.11.1
- pexpect: 4.8.0
- pickleshare: 0.7.5
- pillow: 9.2.0
- pip: 21.1.1
- pipreqs: 0.4.11
- pkgutil-resolve-name: 1.3.10
- platformdirs: 2.6.2
- plot-keras-history: 1.1.38
- plotly: 5.13.1
- pox: 0.3.2
- ppft: 1.7.6.6
- prettytable: 3.6.0
- prometheus-client: 0.15.0
- prompt-toolkit: 3.0.31
- protobuf: 3.19.6
- psutil: 5.9.3
- ptyprocess: 0.7.0
- pure-eval: 0.2.2
- py: 1.11.0
- pyarrow: 6.0.1
- pyasn1: 0.4.8
- pyasn1-modules: 0.2.8
- pycparser: 2.21
- pycrs: 1.0.2
- pydantic: 1.10.7
- pygments: 2.13.0
- pyjwt: 2.6.0
- pyparsing: 3.0.9
- pyperclip: 1.8.2
- pyproj: 3.4.0
- pyrsistent: 0.18.1
- pyshp: 2.3.1
- pysocks: 1.7.1
- python-box: 6.0.2
- python-dateutil: 2.8.2
- python-dotenv: 1.0.0
- python-editor: 1.0.4
- python-json-logger: 2.0.4
- python-multipart: 0.0.6
- pytorch-lightning: 2.0.0
- pytz: 2022.5
- pyyaml: 6.0
- pyzmq: 24.0.1
- qtconsole: 5.4.0
- qtpy: 2.3.0
- rasterio: 1.3.5
- ratelim: 0.1.6
- readchar: 4.0.5
- requests: 2.28.1
- requests-oauthlib: 1.3.1
- requests-toolbelt: 0.10.1
- restee: 0.0.3
- retry: 0.9.2
- retrying: 1.3.3
- rfc3339-validator: 0.1.4
- rfc3986: 1.5.0
- rfc3986-validator: 0.1.1
- rich: 13.3.3
- rsa: 4.9
- ruamel.yaml: 0.17.21
- ruamel.yaml.clib: 0.2.7
- sanitize-ml-labels: 1.0.50
- sankee: 0.2.0
- scikeras: 0.10.0
- scikit-learn: 1.1.3
- scipy: 1.9.3
- scooby: 0.7.0
- seaborn: 0.12.2
- send2trash: 1.8.0
- setuptools: 56.0.0
- shapely: 1.8.5.post1
- six: 1.16.0
- sklearn-pandas: 2.2.0
- sniffio: 1.3.0
- snuggs: 1.4.7
- sortedcontainers: 2.4.0
- soupsieve: 2.3.2.post1
- sqlalchemy: 2.0.2
- stack-data: 0.5.1
- starlette: 0.22.0
- starsessions: 1.3.0
- stevedore: 5.0.0
- support-developer: 1.0.5
- sympy: 1.11.1
- tb-nightly: 2.13.0a20230412
- tblib: 1.7.0
- tenacity: 8.1.0
- tensorboard: 2.11.2
- tensorboard-data-server: 0.6.1
- tensorboard-plugin-wit: 1.8.1
- tensorflow: 2.11.0
- tensorflow-estimator: 2.11.0
- tensorflow-io-gcs-filesystem: 0.28.0
- termcolor: 2.0.1
- terminado: 0.17.0
- testpath: 0.6.0
- threadpoolctl: 3.1.0
- tinycss2: 1.2.1
- tomli: 2.0.1
- toolz: 0.12.0
- torch: 2.0.0
- torch-tb-profiler: 0.4.1
- torchmetrics: 0.11.4
- torchsummary: 1.5.1
- torchview: 0.2.6
- torchvision: 0.15.1
- tornado: 6.2
- tqdm: 4.64.1
- traitlets: 5.5.0
- traittypes: 0.2.1
- triton: 2.0.0
- typing-extensions: 4.4.0
- ujson: 5.7.0
- uri-template: 1.2.0
- uritemplate: 3.0.1
- urllib3: 1.26.12
- uvicorn: 0.21.1
- uvloop: 0.17.0
- watchfiles: 0.19.0
- wcwidth: 0.2.5
- webcolors: 1.12
- webencodings: 0.5.1
- websocket-client: 1.4.1
- websockets: 10.4
- werkzeug: 2.2.2
- wheel: 0.38.4
- whitebox: 2.2.0
- whiteboxgui: 2.2.0
- widgetsnbextension: 3.5.2
- wrapt: 1.14.1
- wxee: 0.3.3
- xarray: 2022.11.0
- xgboost: 1.7.5
- xyzservices: 2022.9.0
- y-py: 0.5.5
- yarg: 0.1.9
- yarl: 1.8.2
- ypy-websocket: 0.8.2
- zarr: 2.10.3
- zict: 2.2.0
- zipp: 3.10.0
* System:
- OS: Linux
- architecture:
- 64bit
- ELF
- processor: x86_64
- python: 3.8.10
- version: #1 SMP Fri Jan 27 02:56:13 UTC 2023
```

### More info

_No response_

cc @carmocca @awaelchli @borda

Reproduce the multiple-inheritance example, then trace initialization through lightning/pytorch/core/datamodule.py, hooks.py, and mixins/hparams_mixin.py as shown in the traceback. Add a regression test for CustomDataModule and verify that it initializes without the reported TypeError.

Written by the indexing model from the issue text.