### 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)

session = dag_maker.session
    ds1_id = session.query(DatasetModel.id).filter_by(uri=dataset1.uri).scalar()
    session.bulk_save_objects(
        [
            DatasetDagRunQueue(dataset_id=ds1_id, target_dag_id=dag2.dag_id),
            DatasetDagRunQueue(dataset_id=ds1_id, target_dag_id=dag3.dag_id),
        ]
    )
    session.flush()
    datasets = session.query(DatasetModel.uri).order_by(DatasetModel.id).all()
    info = get_dataset_triggered_next_run_info([dag1.dag_id], session=session)
    assert info[dag1.dag_id] == {
        "ready": 0,
        "total": 1,
        "uri": datasets[0].uri,
    }
    
    info = get_dataset_triggered_next_run_info([dag2.dag_id, dag3.dag_id], session=session)
    assert info[dag2.dag_id] == {
        "ready": 1,
        "total": 2,
        "uri": "",
    }
    assert info[dag3.dag_id] == {
        "ready": 1,
        "total": 3,
        "uri": "",
    }
def test_dag_uses_timetable_for_run_id(session):
    class CustomRunIdTimetable(Timetable):