### 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)

clear_db_dags()
    clear_db_serialized_dags()
    yield
    clear_db_dags()
    clear_db_serialized_dags()
@pytest.fixture
def clear_datasets():
    clear_db_datasets()
    yield
    clear_db_datasets()
class TestDag:
    def setup_method(self) -> None:
        clear_db_runs()
        clear_db_dags()
        clear_db_datasets()
        self.patcher_dag_code = mock.patch("airflow.models.dag.DagCode.bulk_sync_to_db")
        self.patcher_dag_code.start()
    def teardown_method(self) -> None:
        clear_db_runs()
        clear_db_dags()
        clear_db_datasets()
        self.patcher_dag_code.stop()
    @staticmethod
    def _clean_up(dag_id: str):