### 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)

"""
        Tests if fractional seconds are stored in the database
        """
        dag_id = "test_fractional_seconds"
        dag = DAG(dag_id=dag_id, schedule="@once")
        dag.add_task(BaseOperator(task_id="faketastic", owner="Also fake", start_date=TEST_DATE))
        start_date = timezone.utcnow()
        run = dag.create_dagrun(
            run_id="test_" + start_date.isoformat(),
            execution_date=start_date,
            start_date=start_date,
            state=State.RUNNING,
            external_trigger=False,
        )
        run.refresh_from_db()
        assert start_date == run.execution_date, "dag run execution_date loses precision"
        assert start_date == run.start_date, "dag run start_date loses precision "
        self._clean_up(dag_id)
    def test_pickling(self):
        test_dag_id = "test_pickling"
        args = {"owner": "airflow", "start_date": DEFAULT_DATE}
        dag = DAG(test_dag_id, default_args=args)
        dag_pickle = dag.pickle()
        assert dag_pickle.pickle.dag_id == dag.dag_id
    def test_rich_comparison_ops(self):
        test_dag_id = "test_rich_comparison_ops"
        class DAGsubclass(DAG):