### 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)

"""
        Test scheduling a dag where there is a prior DagRun
        which has the same run_id as the next run should have
        """
        delta = datetime.timedelta(hours=1)
        dag_id = "test_schedule_dag_fake_scheduled_previous"
        dag = DAG(dag_id=dag_id, schedule=delta, start_date=DEFAULT_DATE)
        dag.add_task(BaseOperator(task_id="faketastic", owner="Also fake", start_date=DEFAULT_DATE))
        dag.create_dagrun(
            run_type=DagRunType.SCHEDULED,
            execution_date=DEFAULT_DATE,
            state=State.SUCCESS,
            external_trigger=True,
        )
        dag.sync_to_db()
        with create_session() as session:
            model = session.get(DagModel, dag.dag_id)
        
        
        assert model.next_dagrun == DEFAULT_DATE
        assert model.next_dagrun_create_after == DEFAULT_DATE + delta
        self._clean_up(dag_id)
    def test_schedule_dag_once(self):