### Apache Airflow version

2.6.3

### What happened

When running multiple standalone DAG processors with separate DAG directories using the `--subdir` argument the processors keep deactivating each other's DAGs (and reactivating their own).

After stepping through the code with a debugger I think the issue is that the calls [here](https://github.com/apache/airflow/blob/2.6.3/airflow/dag_processing/manager.py#L794) and [here](https://github.com/apache/airflow/blob/2.6.3/airflow/dag_processing/manager.py#L798) have no awareness of the DAG directories.

### What you think should happen instead

The DAG processors should not touch each other's DAGs in the metadata DB.

### How to reproduce

Start two or more standalone DAG processors with separate DAG directories and observe (e.g. via the UI) how the list of active DAGs keeps changing constantly.

### Operating System

Linux 94b223524983 6.1.32-0-virt #1-Alpine SMP PREEMPT_DYNAMIC Mon, 05 Jun 2023 09:39:09 +0000 x86_64 x86_64 x86_64 GNU/Linux

### Versions of Apache Airflow Providers

_No response_

### Deployment

Other Docker-based deployment

### Deployment details

_No response_

### Anything else

_No response_

### Are you willing to submit PR?

- [ ] Yes I am willing to submit a PR!

### Code of Conduct

- [X] I agree to follow this project's [Code of Conduct](https://github.com/apache/airflow/blob/main/CODE_OF_CONDUCT.md)

mock_pid.return_value = 1234
        mock_waitable_handle.return_value = 3
        manager = DagProcessorJobRunner(
            job=Job(),
            processor=DagFileProcessorManager(
                dag_directory="directory",
                max_runs=1,
                processor_timeout=timedelta(seconds=5),
                signal_conn=MagicMock(),
                dag_ids=[],
                pickle_dags=False,
                async_mode=True,
            ),
        )
        processor = DagFileProcessorProcess(
            file_path="abc.txt",
            pickle_dags=False,
            dag_ids=[],
            dag_directory=TEST_DAG_FOLDER,
            callback_requests=[],
        )
        processor._start_time = timezone.make_aware(datetime.min)
        manager.processor._processors = {"abc.txt": processor}
        manager.processor.waitables[3] = processor
        initial_waitables = len(manager.processor.waitables)
        manager.processor._kill_timed_out_processors()
        mock_kill.assert_called_once_with()
        assert len(manager.processor._processors) == 0
        assert len(manager.processor.waitables) == initial_waitables - 1