### 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)

"""
        We check that has_import_error is false for dags
        being set to scheduler to create dagruns
        """
        dag = DAG(dag_id="test_dags", start_date=DEFAULT_DATE)
        EmptyOperator(task_id="dummy", dag=dag, owner="airflow")
        orm_dag = DagModel(
            dag_id=dag.dag_id,
            has_task_concurrency_limits=False,
            next_dagrun=DEFAULT_DATE,
            next_dagrun_create_after=DEFAULT_DATE + timedelta(days=1),
            is_active=True,
        )
        assert not orm_dag.has_import_errors
        session.add(orm_dag)
        session.flush()
        query, _ = DagModel.dags_needing_dagruns(session)