save_hyperparameter incorrectly infers parameters from superclass

@awaelchli is already working on this.

Since Mar 8, 2024.

### Bug description

Given a model with a submodel, both of which call `save_hyperparameters`, hyperparameters of the submodel that share a name with the main model are overwritten.

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

v2.2

### How to reproduce the bug

```python
from pytorch_lightning.core.mixins.hparams_mixin import HyperparametersMixin

class Submodel(HyperparametersMixin):
def __init__(self, hparam: int):
super().__init__()
self.save_hyperparameters()

class Model(HyperparametersMixin):
def __init__(self, hparam: int):
super().__init__()
self.submodel = Submodel(a=3)
self.save_hyperparameters()

model = Model(hparam=5)
print(model.hparams)
print(model.submodel.hparams)
```

**Expectation**: `model.hparams.hparam == 5`, `model.submodel.hparams.hparam == 3`
**Reality: `model.hparams.hparam == 5`, `model.submodel.hparams.hparam == 5`

### Error messages and logs

_No response_

### Environment

Current environment

* CUDA:
- GPU: None
- available: False
- version: None
* Lightning:
- lightning: 2.2.1
- lightning-utilities: 0.8.0
- pytorch-lightning: 2.2.1
- torch: 2.0.1
- torch-cluster: 1.6.1
- torch-geometric: 2.3.1
- torchmetrics: 1.0.0
* Packages:
- accessible-pygments: 0.0.4
- aiohttp: 3.8.4
- aiosignal: 1.3.1
- alabaster: 0.7.13
- alembic: 1.11.1
- antlr4-python3-runtime: 4.9.3
- anyio: 3.7.0
- appdirs: 1.4.4
- appnope: 0.1.3
- argon2-cffi: 21.3.0
- argon2-cffi-bindings: 21.2.0
- arrow: 1.2.3
- astroid: 3.0.0
- asttokens: 2.2.1
- async-lru: 2.0.2
- async-timeout: 4.0.2
- attrs: 23.1.0
- awkward: 2.2.4
- awkward-cpp: 17
- babel: 2.12.1
- backcall: 0.2.0
- backports.functools-lru-cache: 1.6.4
- beautifulsoup4: 4.12.2
- bleach: 6.0.0
- certifi: 2023.5.7
- cffi: 1.15.1
- cfgv: 3.3.1
- charset-normalizer: 3.1.0
- click: 8.1.3
- cmaes: 0.9.1
- codespell: 2.2.6
- colorama: 0.4.6
- colorlog: 6.7.0
- comm: 0.1.3
- commonmark: 0.9.1
- contourpy: 1.0.7
- coolname: 2.2.0
- coverage: 7.2.7
- cycler: 0.11.0
- debugpy: 1.6.7
- decorator: 5.1.1
- defusedxml: 0.7.1
- dill: 0.3.7
- diskcache: 5.6.3
- distlib: 0.3.6
- docker-pycreds: 0.4.0
- docstring-parser: 0.15
- docutils: 0.19
- entrypoints: 0.4
- exceptiongroup: 1.1.1
- executing: 1.2.0
- fastjsonschema: 2.17.1
- filelock: 3.12.0
- flit-core: 3.9.0
- fonttools: 4.39.4
- fqdn: 1.5.1
- frozenlist: 1.3.3
- fsspec: 2023.6.0
- gitdb: 4.0.10
- gitpython: 3.1.31
- gmpy2: 2.1.2
- gnn-tracking: 0.0.1
- gnntrack: 0.0.1
- greenlet: 2.0.2
- hpo2: 0.1.0
- hsfparana: 0.1.0
- hydra-core: 1.3.2
- identify: 2.5.24
- idna: 3.4
- imagesize: 1.4.1
- importlib-metadata: 6.6.0
- importlib-resources: 5.12.0
- iniconfig: 2.0.0
- ipykernel: 6.23.1
- ipython: 8.14.0
- ipython-genutils: 0.2.0
- ipywidgets: 8.0.6
- isoduration: 20.11.0
- isort: 5.12.0
- jedi: 0.18.2
- jinja2: 3.1.2
- joblib: 1.2.0
- json5: 0.9.14
- jsonargparse: 4.21.2
- jsonpointer: 2.4
- jsonschema: 4.17.3
- jupyter: 1.0.0
- jupyter-client: 8.2.0
- jupyter-console: 6.6.3
- jupyter-core: 5.3.0
- jupyter-events: 0.6.3
- jupyter-lsp: 2.2.0
- jupyter-server: 2.6.0
- jupyter-server-terminals: 0.4.4
- jupyterlab: 4.0.2
- jupyterlab-pygments: 0.2.2
- jupyterlab-server: 2.23.0
- jupyterlab-widgets: 3.0.7
- kiwisolver: 1.4.4
- lazy-object-proxy: 1.9.0
- lightning: 2.2.1
- lightning-utilities: 0.8.0
- llvmlite: 0.40.1
- mako: 1.2.4
- markdown-it-py: 3.0.0
- markupsafe: 2.1.3
- matplotlib: 3.7.1
- matplotlib-inline: 0.1.6
- mccabe: 0.7.0
- mdmm: 0.1.3
- mdurl: 0.1.2
- mistune: 2.0.5
- mplhep: 0.3.28
- mplhep-data: 0.0.3
- mpmath: 1.3.0
- msgpack: 1.0.5
- multidict: 6.0.4
- nbclassic: 1.0.0
- nbclient: 0.8.0
- nbconvert: 7.4.0
- nbformat: 5.9.0
- nest-asyncio: 1.5.6
- networkx: 3.1
- nodeenv: 1.8.0
- notebook: 6.5.4
- notebook-shim: 0.2.3
- numba: 0.57.1
- numpy: 1.24.4
- object-condensation: 0.1.dev20+gf5708c7
- omegaconf: 2.3.0
- optuna: 3.2.0
- overrides: 7.3.1
- packaging: 23.1
- pandas: 2.0.2
- pandocfilters: 1.5.0
- parso: 0.8.3
- pathtools: 0.1.2
- pexpect: 4.8.0
- pickleshare: 0.7.5
- pillow: 9.5.0
- pip: 23.1.2
- pkgutil-resolve-name: 1.3.10
- platformdirs: 3.5.1
- pluggy: 1.0.0
- pooch: 1.7.0
- pre-commit: 3.3.2
- prometheus-client: 0.17.0
- prompt-toolkit: 3.0.38
- protobuf: 3.20.3
- psutil: 5.9.5
- ptyprocess: 0.7.0
- pure-eval: 0.2.2
- pyarrow: 12.0.1
- pycparser: 2.21
- pydata-sphinx-theme: 0.13.3
- pygments: 2.15.1
- pylint: 3.0.0
- pyobjc-core: 9.2
- pyobjc-framework-cocoa: 9.2
- pyparsing: 3.0.9
- pyrsistent: 0.19.3
- pysocks: 1.7.1
- pytest: 7.4.0
- pytest-cov: 4.1.0
- pytest-cover: 3.0.0
- pytest-coverage: 0.0
- python-dateutil: 2.8.2
- python-frontmatter: 1.0.0
- python-json-logger: 2.0.7
- pytorch-lightning: 2.2.1
- pytz: 2023.3
- pyyaml: 6.0
- pyzmq: 25.1.0
- ray: 2.5.1
- recommonmark: 0.7.1
- requests: 2.31.0
- rfc3339-validator: 0.1.4
- rfc3986-validator: 0.1.1
- rich: 13.4.2
- ruff: 0.0.276
- scienceplots: 2.1.1
- scikit-learn: 1.2.2
- scipy: 1.10.1
- send2trash: 1.8.2
- sentry-sdk: 1.21.1
- setproctitle: 1.3.2
- setuptools: 67.7.2
- six: 1.16.0
- smmap: 3.0.5
- sniffio: 1.3.0
- snowballstemmer: 2.2.0
- soupsieve: 2.3.2.post1
- sphinx: 6.2.1
- sphinx-autoapi: 2.1.0
- sphinx-book-theme: 1.0.1
- sphinxcontrib-applehelp: 1.0.4
- sphinxcontrib-devhelp: 1.0.2
- sphinxcontrib-htmlhelp: 2.0.1
- sphinxcontrib-jsmath: 1.0.1
- sphinxcontrib-qthelp: 1.0.3
- sphinxcontrib-serializinghtml: 1.1.5
- sqlalchemy: 2.0.15
- stack-data: 0.6.2
- sympy: 1.12
- tabulate: 0.9.0
- tensorboardx: 2.6
- terminado: 0.17.1
- threadpoolctl: 3.1.0
- tinycss2: 1.2.1
- tomlkit: 0.12.1
- torch: 2.0.1
- torch-cluster: 1.6.1
- torch-geometric: 2.3.1
- torchmetrics: 1.0.0
- tornado: 6.3.2
- tqdm: 4.65.0
- trackml: 3
- traitlets: 5.9.0
- types-markupsafe: 1.1.10
- typeshed-client: 2.3.0
- typing-extensions: 4.6.3
- typing-utils: 0.1.0
- tzdata: 2023.3
- uhi: 0.3.3
- unidecode: 1.3.6
- uproot: 5.0.9
- uri-template: 1.3.0
- urllib3: 2.0.3
- virtualenv: 20.23.0
- wandb: 0.16.3
- wandb-osh: 1.0.4
- wcwidth: 0.2.6
- webcolors: 1.13
- webencodings: 0.5.1
- websocket-client: 1.5.2
- wheel: 0.40.0
- widgetsnbextension: 4.0.7
- wrapt: 1.15.0
- yarl: 1.9.2
- zipp: 3.15.0
* System:
- OS: Darwin
- architecture:
- 64bit
-
- processor: arm
- python: 3.11.3
- release: 23.2.0
- version: Darwin Kernel Version 23.2.0: Wed Nov 15 21:53:18 PST 2023; root:xnu-10002.61.3~2/RELEASE_ARM64_T6000

### More info

_No response_