### 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)

existing_dags.get(dag_id)
                and existing_dags.get(dag_id).schedule_dataset_references  
                or []
            )
            dag_refs_to_add = {x for x in dag_refs_needed if x not in dag_refs_stored}
            session.bulk_save_objects(dag_refs_to_add)
            for obj in dag_refs_stored - dag_refs_needed:
                session.delete(obj)
        existing_task_outlet_refs_dict = collections.defaultdict(set)
        for dag_id, orm_dag in existing_dags.items():
            for todr in orm_dag.task_outlet_dataset_references:
                existing_task_outlet_refs_dict[(dag_id, todr.task_id)].add(todr)
        
        for (dag_id, task_id), uri_list in outlet_references.items():
            task_refs_needed = {
                TaskOutletDatasetReference(dataset_id=stored_datasets[uri].id, dag_id=dag_id, task_id=task_id)
                for uri in uri_list
            }
            task_refs_stored = existing_task_outlet_refs_dict[(dag_id, task_id)]
            task_refs_to_add = {x for x in task_refs_needed if x not in task_refs_stored}
            session.bulk_save_objects(task_refs_to_add)
            for obj in task_refs_stored - task_refs_needed:
                session.delete(obj)
        
        
        session.flush()
        for dag in dags:
            cls.bulk_write_to_db(dag.subdags, processor_subdir=processor_subdir, session=session)
    @provide_session
    def sync_to_db(self, processor_subdir: str | None = None, session=NEW_SESSION):