### 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)

if not dag_bag:
                        from airflow.models.dagbag import DagBag
                        dag_bag = DagBag(read_dags_from_db=True)
                    external_dag = dag_bag.get_dag(tii.dag_id, session=session)
                    if not external_dag:
                        raise AirflowException(f"Could not find dag {tii.dag_id}")
                    downstream = external_dag.partial_subset(
                        task_ids_or_regex=[tii.task_id],
                        include_upstream=False,
                        include_downstream=True,
                    )
                    result.update(
                        downstream._get_task_instances(
                            task_ids=None,
                            run_id=tii.run_id,
                            start_date=None,
                            end_date=None,
                            state=state,
                            include_subdags=include_subdags,
                            include_dependent_dags=include_dependent_dags,
                            include_parentdag=False,
                            as_pk_tuple=True,
                            exclude_task_ids=exclude_task_ids,
                            dag_bag=dag_bag,
                            session=session,
                            recursion_depth=recursion_depth + 1,
                            max_recursion_depth=max_recursion_depth,
                            visited_external_tis=visited_external_tis,
                        )
                    )