### 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)

"""
            Create a subdag.
            """
            dag_subdag = DAG(
                dag_id=f"{parent_dag_name}.{child_dag_name}",
                schedule="@daily",
                default_args=args,
            )
            for i in range(2):
                EmptyOperator(task_id=f"{child_dag_name}-task-{i + 1}", dag=dag_subdag)
            return dag_subdag
        with DAG(
            dag_id="test_subdag_operator",
            start_date=datetime.datetime(2019, 1, 1),
            max_active_runs=1,
            schedule=timedelta(minutes=1),
        ) as dag:
            section_1 = SubDagOperator(
                task_id="section-1",
                subdag=subdag(dag.dag_id, "section-1", {"start_date": dag.start_date}),
            )
        subdag = section_1.subdag
        
        subdag.parent_dag = dag
        next_parent_info = dag.next_dagrun_info(None)
        assert next_parent_info.logical_date == timezone.datetime(2019, 1, 1, 0, 0)
        next_subdag_info = subdag.next_dagrun_info(None)
        assert next_subdag_info is None, "SubDags should never have DagRuns created by the scheduler"
    def test_replace_outdated_access_control_actions(self):