### 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)

dag.clear()
        self._clean_up(dag_id)
    @patch("airflow.models.dag.Stats")
    def test_dag_handle_callback_crash(self, mock_stats):
        """
        Tests avoid crashes from calling dag callbacks exceptions
        """
        dag_id = "test_dag_callback_crash"
        mock_callback_with_exception = mock.MagicMock()
        mock_callback_with_exception.side_effect = Exception
        dag = DAG(
            dag_id=dag_id,
            
            on_success_callback=lambda: 1,
            on_failure_callback=mock_callback_with_exception,
        )
        when = TEST_DATE
        dag.add_task(BaseOperator(task_id="faketastic", owner="Also fake", start_date=when))
        with create_session() as session:
            dag_run = dag.create_dagrun(State.RUNNING, when, run_type=DagRunType.MANUAL, session=session)
            
            dag.handle_callback(dag_run, success=False)
            dag.handle_callback(dag_run, success=True)
        mock_stats.incr.assert_called_with(
            "dag.callback_exceptions",
            tags={"dag_id": "test_dag_callback_crash"},
        )
        dag.clear()
        self._clean_up(dag_id)
    def test_next_dagrun_after_fake_scheduled_previous(self):