### 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)

assert hash(dag) == hash(dag)
        assert hash(dag_eq) == hash(dag)
        assert hash(dag_diff_name) != hash(dag)
        assert hash(dag_subclass) != hash(dag)
    def test_get_paused_dag_ids(self):
        dag_id = "test_get_paused_dag_ids"
        dag = DAG(dag_id, is_paused_upon_creation=True)
        dag.sync_to_db()
        assert DagModel.get_dagmodel(dag_id) is not None
        paused_dag_ids = DagModel.get_paused_dag_ids([dag_id])
        assert paused_dag_ids == {dag_id}
        with create_session() as session:
            session.query(DagModel).filter(DagModel.dag_id == dag_id).delete(synchronize_session=False)
    @pytest.mark.parametrize(
        "schedule_interval_arg, expected_timetable, interval_description",
        [
            (None, NullTimetable(), "Never, external triggers only"),
            ("@daily", cron_timetable("0 0 * * *"), "At 00:00"),
            ("@weekly", cron_timetable("0 0 * * 0"), "At 00:00, only on Sunday"),
            ("@monthly", cron_timetable("0 0 1 * *"), "At 00:00, on day 1 of the month"),
            ("@quarterly", cron_timetable("0 0 1 */3 *"), "At 00:00, on day 1 of the month, every 3 months"),
            ("@yearly", cron_timetable("0 0 1 1 *"), "At 00:00, on day 1 of the month, only in January"),
            ("5 0 * 8 *", cron_timetable("5 0 * 8 *"), "At 00:05, only in August"),
            ("@once", OnceTimetable(), "Once, as soon as possible"),
            (datetime.timedelta(days=1), delta_timetable(datetime.timedelta(days=1)), ""),
            ("30 21 * * 5 1", cron_timetable("30 21 * * 5 1"), ""),
        ],
    )
    def test_timetable_and_description_from_schedule_interval_arg(