### 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)

dag_id = "test_clear_set_dagrun_state"
        self._clean_up(dag_id)
        task_id = "t1"
        dag = DAG(dag_id, start_date=DEFAULT_DATE, max_active_runs=1)
        @dag.task
        def make_arg_lists():
            return [[1], [2], [{"a": "b"}]]
        def consumer(value):
            print(value)
        mapped = PythonOperator.partial(task_id=task_id, dag=dag, python_callable=consumer).expand(
            op_args=make_arg_lists()
        )
        session = settings.Session()
        dagrun_1 = dag.create_dagrun(
            run_type=DagRunType.BACKFILL_JOB,
            state=State.FAILED,
            start_date=DEFAULT_DATE,
            execution_date=DEFAULT_DATE,
            session=session,
        )
        expand_mapped_task(mapped, dagrun_1.run_id, "make_arg_lists", length=2, session=session)
        upstream_ti = dagrun_1.get_task_instance("make_arg_lists", session=session)
        ti = dagrun_1.get_task_instance(task_id, map_index=0, session=session)
        ti2 = dagrun_1.get_task_instance(task_id, map_index=1, session=session)
        upstream_ti.state = State.SUCCESS
        ti.state = State.SUCCESS