### 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)

DatasetDagRunQueue(dataset_id=ds1_id, target_dag_id=dag.dag_id, created_at=DEFAULT_DATE),
                DatasetDagRunQueue(
                    dataset_id=ds2_id, target_dag_id=dag.dag_id, created_at=DEFAULT_DATE + timedelta(hours=1)
                ),
            ]
        )
        session.flush()
        query, dataset_triggered_dag_info = DagModel.dags_needing_dagruns(session)
        assert 1 == len(dataset_triggered_dag_info)
        assert dag.dag_id in dataset_triggered_dag_info
        first_queued_time, last_queued_time = dataset_triggered_dag_info[dag.dag_id]
        assert first_queued_time == DEFAULT_DATE
        assert last_queued_time == DEFAULT_DATE + timedelta(hours=1)
class TestQueries:
    def setup_method(self) -> None:
        clear_db_runs()
    def teardown_method(self) -> None:
        clear_db_runs()
    @pytest.mark.parametrize("tasks_count", [3, 12])
    def test_count_number_queries(self, tasks_count):
        dag = DAG("test_dagrun_query_count", start_date=DEFAULT_DATE)
        for i in range(tasks_count):
            EmptyOperator(task_id=f"dummy_task_{i}", owner="test", dag=dag)
        with assert_queries_count(2):
            dag.create_dagrun(
                run_id="test_dagrun_query_count",
                state=State.RUNNING,
                execution_date=TEST_DATE,
            )
class TestDagDecorator: