### 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)

manager = DagProcessorJobRunner(
            job=Job(),
            processor=DagFileProcessorManager(
                dag_directory=TEST_DAG_FOLDER,
                max_runs=1,
                processor_timeout=timedelta(days=365),
                signal_conn=MagicMock(),
                dag_ids=[],
                pickle_dags=False,
                async_mode=True,
            ),
        )
        dag1_req1 = DagCallbackRequest(
            full_filepath="/green_eggs/ham/file1.py",
            dag_id="dag1",
            run_id="run1",
            is_failure_callback=False,
            processor_subdir=tmpdir,
            msg=None,
        )
        dag1_req2 = DagCallbackRequest(
            full_filepath="/green_eggs/ham/file1.py",
            dag_id="dag1",
            run_id="run1",
            is_failure_callback=False,
            processor_subdir=tmpdir,
            msg=None,