### 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)

DagRunInfo.interval(
                    DEFAULT_DATE - datetime.timedelta(hours=2),
                    DEFAULT_DATE - datetime.timedelta(hours=1),
                ),
                DagRunInfo.interval(
                    DEFAULT_DATE - datetime.timedelta(hours=1),
                    DEFAULT_DATE,
                ),
                DagRunInfo.interval(
                    DEFAULT_DATE,
                    DEFAULT_DATE + datetime.timedelta(hours=1),
                ),
            ],
        ),
    ],
    ids=["in-dag-restriction", "out-of-dag-restriction"],
)
def test_iter_dagrun_infos_between(start_date, expected_infos):
    dag = DAG(dag_id="test_get_dates", start_date=DEFAULT_DATE, schedule="@hourly")
    EmptyOperator(task_id="dummy", dag=dag)
    iterator = dag.iter_dagrun_infos_between(
        earliest=pendulum.instance(start_date),
        latest=pendulum.instance(DEFAULT_DATE),
        align=True,
    )
    assert expected_infos == list(iterator)
def test_iter_dagrun_infos_between_error(caplog):
    start = pendulum.instance(DEFAULT_DATE - datetime.timedelta(hours=1))
    end = pendulum.instance(DEFAULT_DATE)
    class FailingAfterOneTimetable(Timetable):