### 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)

dag = dag_by_ids[missing_dag_id]
            if dag.is_paused_upon_creation is not None:
                orm_dag.is_paused = dag.is_paused_upon_creation
            orm_dag.tags = []
            log.info("Creating ORM DAG for %s", dag.dag_id)
            session.add(orm_dag)
            orm_dags.append(orm_dag)
        most_recent_runs: dict[str, DagRun] = {}
        num_active_runs: dict[str, int] = {}
        
        if any(dag.timetable.can_be_scheduled for dag in dags):
            
            most_recent_subq = (
                select(DagRun.dag_id, func.max(DagRun.execution_date).label("max_execution_date"))
                .where(
                    DagRun.dag_id.in_(existing_dags),
                    or_(DagRun.run_type == DagRunType.BACKFILL_JOB, DagRun.run_type == DagRunType.SCHEDULED),
                )
                .group_by(DagRun.dag_id)
                .subquery()
            )
            most_recent_runs_iter = session.scalars(
                select(DagRun).where(
                    DagRun.dag_id == most_recent_subq.c.dag_id,
                    DagRun.execution_date == most_recent_subq.c.max_execution_date,
                )
            )
            most_recent_runs = {run.dag_id: run for run in most_recent_runs_iter}
            
            num_active_runs = DagRun.active_runs_of_dags(dag_ids=existing_dags, session=session)