### 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_run_state=False,  
            session=session,
        )
        self.log.debug("Getting dagrun for dag %s", self.dag_id)
        logical_date = timezone.coerce_datetime(execution_date)
        data_interval = self.timetable.infer_manual_data_interval(run_after=logical_date)
        dr: DagRun = _get_or_create_dagrun(
            dag=self,
            start_date=execution_date,
            execution_date=execution_date,
            run_id=DagRun.generate_run_id(DagRunType.MANUAL, execution_date),
            session=session,
            conf=run_conf,
            data_interval=data_interval,
        )
        tasks = self.task_dict
        self.log.debug("starting dagrun")
        
        
        
        while dr.state == DagRunState.RUNNING:
            schedulable_tis, _ = dr.update_state(session=session)
            for ti in schedulable_tis:
                try:
                    add_logger_if_needed(ti)
                    ti.task = tasks[ti.task_id]
                    _run_task(ti, session=session)
                except Exception:
                    self.log.exception("Task failed; ti=%s", ti)
        if conn_file_path or variable_file_path: