### 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)

filelocs = []
        for orm_dag in sorted(orm_dags, key=lambda d: d.dag_id):
            dag = dag_by_ids[orm_dag.dag_id]
            filelocs.append(dag.fileloc)
            if dag.is_subdag:
                orm_dag.is_subdag = True
                orm_dag.fileloc = dag.parent_dag.fileloc  
                orm_dag.root_dag_id = dag.parent_dag.dag_id  
                orm_dag.owners = dag.parent_dag.owner  
            else:
                orm_dag.is_subdag = False
                orm_dag.fileloc = dag.fileloc
                orm_dag.owners = dag.owner
            orm_dag.is_active = True
            orm_dag.has_import_errors = False
            orm_dag.last_parsed_time = timezone.utcnow()
            orm_dag.default_view = dag.default_view
            orm_dag.description = dag.description
            orm_dag.max_active_tasks = dag.max_active_tasks
            orm_dag.max_active_runs = dag.max_active_runs
            orm_dag.has_task_concurrency_limits = any(
                t.max_active_tis_per_dag is not None or t.max_active_tis_per_dagrun is not None
                for t in dag.tasks
            )
            orm_dag.schedule_interval = dag.schedule_interval
            orm_dag.timetable_description = dag.timetable.description
            orm_dag.processor_subdir = processor_subdir
            run: DagRun | None = most_recent_runs.get(dag.dag_id)
            if run is None:
                data_interval = None