### 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)

dataset = Dataset(uri="hello")
        with dag_maker(
            session=session,
            dag_id="my_dag",
            max_active_runs=1,
            schedule=[dataset],
            start_date=pendulum.now().add(days=-2),
        ) as dag:
            EmptyOperator(task_id="dummy")
        
        query, _ = DagModel.dags_needing_dagruns(session)
        dag_models = query.all()
        assert dag_models == []
        
        dag_model = session.query(DagModel).filter(DagModel.dag_id == dag.dag_id).one()
        dataset_model: DatasetModel = dag_model.schedule_datasets[0]
        session.add(DatasetDagRunQueue(dataset_id=dataset_model.id, target_dag_id=dag_model.dag_id))
        session.flush()
        query, _ = DagModel.dags_needing_dagruns(session)
        dag_models = query.all()
        assert dag_models == [dag_model]
        
        dag_maker.create_dagrun(
            run_type=DagRunType.DATASET_TRIGGERED,
            state=DagRunState.QUEUED,