### 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)

EmptyOperator(task_id=task_id, dag=dag2)
        DAG.bulk_write_to_db([dag1, dag2], session=session)
        session.commit()
        session.expunge_all()
        stored_datasets = {x.uri: x for x in session.query(DatasetModel).all()}
        d1_orm = stored_datasets[d1.uri]
        d2_orm = stored_datasets[d2.uri]
        assert [x.dag_id for x in d1_orm.consuming_dags] == []
        assert set(
            session.query(
                TaskOutletDatasetReference.task_id,
                TaskOutletDatasetReference.dag_id,
                TaskOutletDatasetReference.dataset_id,
            )
            .filter(TaskOutletDatasetReference.dag_id.in_((dag_id1, dag_id2)))
            .all()
        ) == {(task_id, dag_id1, d2_orm.id)}
    def test_bulk_write_to_db_unorphan_datasets(self):
        """
        Datasets can lose their last reference and be orphaned, but then if a reference to them reappears, we
        need to un-orphan those datasets
        """
        with create_session() as session:
            
            
            dataset1 = Dataset(uri="ds1")
            dataset2 = Dataset(uri="ds2")
            session.add(DatasetModel(uri=dataset2.uri, is_orphaned=True))
            dataset3 = Dataset(uri="ds3")
            dataset4 = Dataset(uri="ds4")