### 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)

with create_session() as session:
            session.add(DagModel(dag_id="dag_default_view_old", default_view=None))
            session.commit()
            orm_dag = session.query(DagModel).filter(DagModel.dag_id == "dag_default_view_old").one()
        assert orm_dag.default_view is None
        assert orm_dag.get_default_view() == conf.get("webserver", "dag_default_view").lower()
    def test_dag_is_deactivated_upon_dagfile_deletion(self):
        dag_id = "old_existing_dag"
        dag_fileloc = "/usr/local/airflow/dags/non_existing_path.py"
        dag = DAG(
            dag_id,
            is_paused_upon_creation=True,
        )
        dag.fileloc = dag_fileloc
        session = settings.Session()
        with mock.patch("airflow.models.dag.DagCode.bulk_sync_to_db"):
            dag.sync_to_db(session=session)
        orm_dag = session.query(DagModel).filter(DagModel.dag_id == dag_id).one()
        assert orm_dag.is_active
        assert orm_dag.fileloc == dag_fileloc
        DagModel.deactivate_deleted_dags(list_py_file_paths(settings.DAGS_FOLDER))
        orm_dag = session.query(DagModel).filter(DagModel.dag_id == dag_id).one()
        assert not orm_dag.is_active
        session.execute(DagModel.__table__.delete().where(DagModel.dag_id == dag_id))
        session.close()
    def test_dag_naive_default_args_start_date_with_timezone(self):