### 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)

"""
        Test `default_view` default value of DAG initialization
        """
        dag = models.DAG(dag_id="test-default_default_view")
        assert conf.get("webserver", "dag_default_view").lower() == dag.default_view
    def test_dag_invalid_orientation(self):
        """
        Test invalid `orientation` of DAG initialization
        """
        with pytest.raises(AirflowException, match="Invalid values of dag.orientation: only support"):
            models.DAG(dag_id="test-invalid-orientation", orientation="airflow")
    def test_dag_orientation_default_value(self):
        """
        Test `orientation` default value of DAG initialization
        """
        dag = models.DAG(dag_id="test-default_orientation")
        assert conf.get("webserver", "dag_orientation") == dag.orientation
    def test_dag_as_context_manager(self):
        """
        Test DAG as a context manager.
        When used as a context manager, Operators are automatically added to
        the DAG (unless they specify a different DAG)
        """
        dag = DAG("dag", start_date=DEFAULT_DATE, default_args={"owner": "owner1"})
        dag2 = DAG("dag2", start_date=DEFAULT_DATE, default_args={"owner": "owner2"})
        with dag:
            op1 = EmptyOperator(task_id="op1")
            op2 = EmptyOperator(task_id="op2", dag=dag2)