MLFlow Logger Tests Not Passing

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### Bug description

Currently, on the continuous integration tests, they are not calling `mlflow` as the tests are skipped as the package isn't included in the development dependencies. [The tests I am referring to are here.](https://github.com/Lightning-AI/pytorch-lightning/blob/28f18dcc4e806f42c0191671a55a1da4fcb57ca2/tests/tests_pytorch/loggers/test_mlflow.py)

This could be fixed by adding `mlflow` to `requirements/pytorch/test.txt`, but doing so would cause the tests to fail. The tests fail for three separate reasons.

### FutureWarning

In `tests/tests_pytorch/loggers/test_mlflow.py`, two tests fail due to future warnings:

* `test_mlflow_run_name_setting`
* `test_mlflow_logger_dirs_creation`

These occur from version 3.6.0, the latest being 3.7.0 at the time of writing.

The warning message linked to [this issue](https://github.com/mlflow/mlflow/issues/18534) and is as follows:
```
FutureWarning: The filesystem tracking backend (e.g., './mlruns') will be deprecated in February 2026. Consider transitioning to a database backend (e.g., 'sqlite:///mlflow.db') to take advantage of the latest MLflow features. See https://github.com/mlflow/mlflow/issues/18534 for more details and migration guidance.
```

### test_mlflow_run_name_setting

If you downgrade MLFlow to `mlflow<=3.5.0`, then the `FutureWarnings` disappear (they can also be suppressed), but the tests still do not pass.

They fail because `test_mlflow_run_name_setting` - Produces an `AssertionError: expected call not found` which relates to the class passing `'mlflow.runName': 'run-name-1'` as an option to the class.

This is because the tests create a `client` and a `logger`, reinitialise the client, but do not attach the logger each time.
This means the assertion on `client` refers to the `logger` in the latter parts of the test, which is not the one being adjusted in the preceding lines.

### restore_env_variables

In the `tests/tests_pytorch/conftest.py`, there is a list of environment variables that are created that should be reset at the end of the tests.
If the `mlflow` package is installed, then the variable `MLFLOW_TRACKING_URI` is expected from the behaviour of MLFlow.

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

master

### Reproduced in studio

_No response_

### How to reproduce the bug

```python
git clone https://github.com/Lightning-AI/pytorch-lightning
cd pytorch-lightning
uv venv -p 3.12
source .venv/bin/activate
make setup
uv pip install "mlflow>=3.6"

Then you can either run `make test` or

pytest tests/tests_pytorch/loggers/test_mlflow.py
```

### Error messages and logs

### FutureWarning

```
src/lightning/pytorch/loggers/mlflow.py:153: in __init__
self._mlflow_client = MlflowClient(tracking_uri)
^^^^^^^^^^^^^^^^^^^^^^^^^^
.venv/lib/python3.12/site-packages/mlflow/tracking/client.py:224: in __init__
self._tracking_client = TrackingServiceClient(final_tracking_uri)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.venv/lib/python3.12/site-packages/mlflow/tracking/_tracking_service/client.py:96: in __init__
self.store
.venv/lib/python3.12/site-packages/mlflow/tracking/_tracking_service/client.py:100: in store
return utils._get_store(self.tracking_uri)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.venv/lib/python3.12/site-packages/mlflow/tracking/_tracking_service/utils.py:253: in _get_store
return _tracking_store_registry.get_store(store_uri, artifact_uri)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.venv/lib/python3.12/site-packages/mlflow/tracking/_tracking_service/registry.py:45: in get_store
return self._get_store_with_resolved_uri(resolved_store_uri, artifact_uri)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.venv/lib/python3.12/site-packages/mlflow/tracking/_tracking_service/registry.py:56: in _get_store_with_resolved_uri
return builder(store_uri=resolved_store_uri, artifact_uri=artifact_uri)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.venv/lib/python3.12/site-packages/mlflow/tracking/_tracking_service/utils.py:177: in _get_file_store
return FileStore(store_uri, store_uri)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

self = , root_directory = 'file:/.../pytest-of-name/pytest-18/test_mlflow_run_name_setting0'
artifact_root_uri = 'file:/.../pytest-18/test_mlflow_run_name_setting0'

def __init__(self, root_directory=None, artifact_root_uri=None):
"""
Create a new FileStore with the given root directory and a given default artifact root URI.
"""
super().__init__()
> warnings.warn(
"The filesystem tracking backend (e.g., './mlruns') will be deprecated in "
"February 2026. Consider transitioning to a database backend (e.g., "
"'sqlite:///mlflow.db') to take advantage of the latest MLflow features. "
"See https://github.com/mlflow/mlflow/issues/18534 for more details and migration "
"guidance.",
FutureWarning,
stacklevel=2,
)
E FutureWarning: The filesystem tracking backend (e.g., './mlruns') will be deprecated in February 2026. Consider transitioning to a database backend (e.g., 'sqlite:///mlflow.db') to take advantage of the latest MLflow features. See https://github.com/mlflow/mlflow/issues/18534 for more details and migration guidance.

.venv/lib/python3.12/site-packages/mlflow/store/tracking/file_store.py:219: FutureWarning

```

### test_mlflow_run_name_setting

```
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

self = , args = ()
kwargs = {'experiment_id': 'exp-id', 'tags': {'mlflow.source.name': '/.../pytorch-lightning/.venv/bin/pytest', 'mlflow.source.type': 'LOCAL', 'mlflow.user': 'name'}}
expected = call(experiment_id='exp-id', tags={'mlflow.user': 'name', 'mlflow.source.name': '/.../pytorch-lightning/.venv/bin/pytest', 'mlflow.source.type': 'LOCAL'})
actual = call(experiment_id='exp-id', tags={'mlflow.user': 'name', 'mlflow.source.name': '/.../pytorch-lightning/.venv/bin/pytest', 'mlflow.source.type': 'LOCAL', 'mlflow.runName': 'run-name-1'})
_error_message = ._error_message at 0x313d20cc0>, cause = None

def assert_called_with(self, /, *args, **kwargs):
"""assert that the last call was made with the specified arguments.

Raises an AssertionError if the args and keyword args passed in are
different to the last call to the mock."""
if self.call_args is None:
expected = self._format_mock_call_signature(args, kwargs)
actual = 'not called.'
error_message = ('expected call not found.\nExpected: %s\n Actual: %s'
% (expected, actual))
raise AssertionError(error_message)

def _error_message():
msg = self._format_mock_failure_message(args, kwargs)
return msg
expected = self._call_matcher(_Call((args, kwargs), two=True))
actual = self._call_matcher(self.call_args)
if actual != expected:
cause = expected if isinstance(expected, Exception) else None
> raise AssertionError(_error_message()) from cause
E AssertionError: expected call not found.
E Expected: create_run(experiment_id='exp-id', tags={'mlflow.user': 'name', 'mlflow.source.name': '/.../pytorch-lightning/.venv/bin/pytest', 'mlflow.source.type': 'LOCAL'})
E Actual: create_run(experiment_id='exp-id', tags={'mlflow.user': 'name', 'mlflow.source.name': '/.../Lightning/pytorch-lightning/.venv/bin/pytest', 'mlflow.source.type': 'LOCAL', 'mlflow.runName': 'run-name-1'})

/.../lib/python3.12/unittest/mock.py:949: AssertionError
```

### restore_env_variables

```
_________________________________________________________________________________ ERROR at teardown of test_mlflow_run_name_setting __________________________________________________________________________________

@pytest.fixture(autouse=True)
def restore_env_variables():
"""Ensures that environment variables set during the test do not leak out."""
env_backup = os.environ.copy()
yield
leaked_vars = os.environ.keys() - env_backup.keys()
# restore environment as it was before running the test
os.environ.clear()
os.environ.update(env_backup)
# these are currently known leakers - ideally these would not be allowed
allowlist = {
"CUBLAS_WORKSPACE_CONFIG", # enabled with deterministic flag
"CUDA_DEVICE_ORDER",
"LOCAL_RANK",
"NODE_RANK",
"WORLD_SIZE",
"MASTER_ADDR",
"MASTER_PORT",
"PL_GLOBAL_SEED",
"PL_SEED_WORKERS",
"WANDB_MODE",
"WANDB_REQUIRE_SERVICE",
"WANDB_SERVICE",
"RANK", # set by DeepSpeed
"CUDA_MODULE_LOADING", # leaked by PyTorch
"KMP_INIT_AT_FORK", # leaked by PyTorch
"KMP_DUPLICATE_LIB_OK", # leaked by PyTorch
"CRC32C_SW_MODE", # leaked by tensorboardX
"TRITON_CACHE_DIR", # leaked by torch.compile
"_TORCHINDUCTOR_PYOBJECT_TENSOR_DATA_PTR", # leaked by torch.compile
"OMP_NUM_THREADS", # set by our launchers
# leaked by XLA
"ALLOW_MULTIPLE_LIBTPU_LOAD",
"GRPC_VERBOSITY",
"TF_CPP_MIN_LOG_LEVEL",
"TF_GRPC_DEFAULT_OPTIONS",
"XLA_FLAGS",
"TORCHINDUCTOR_CACHE_DIR", # leaked by torch.compile
# TensorFlow and TPU related variables
"TF2_BEHAVIOR",
"TPU_ML_PLATFORM",
"TPU_ML_PLATFORM_VERSION",
"LD_LIBRARY_PATH",
"ENABLE_RUNTIME_UPTIME_TELEMETRY",
}
leaked_vars.difference_update(allowlist)
> assert not leaked_vars, f"test is leaking environment variable(s): {set(leaked_vars)}"
E AssertionError: test is leaking environment variable(s): {'MLFLOW_TRACKING_URI'}
E assert not {'MLFLOW_TRACKING_URI'}

tests/tests_pytorch/conftest.py:106: AssertionError

```

### Environment

Current environment

```
#- PyTorch Lightning Version (e.g., 2.5.0): 2.6.0
#- PyTorch Version (e.g., 2.5): 2.9.0
#- Python version (e.g., 3.12): 3.12
#- OS (e.g., Linux): MacOS
#- CUDA/cuDNN version: None
#- GPU models and configuration: None
#- How you installed Lightning(`conda`, `pip`, source): source
```

### More info

_No response_

cc @ethanwharris

Start with requirements/pytorch/test.txt and run pytest tests/tests_pytorch/loggers/test_mlflow.py using the reproduction setup. Read tests/tests_pytorch/loggers/test_mlflow.py and tests/tests_pytorch/conftest.py, focusing on the named failing tests, warning behavior, mock assertions, and MLFLOW_TRACKING_URI cleanup. Done means MLflow is exercised in CI and the logger tests and environment-restoration checks pass.

Written by the indexing model from the issue text.