### 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)

self._processors: dict[str, DagFileProcessorProcess] = {}
        self._num_run = 0
        
        self._file_stats: dict[str, DagFileStat] = {}
        
        self.last_dag_dir_refresh_time = timezone.make_aware(datetime.fromtimestamp(0))
        
        self.last_stat_print_time = 0
        
        self.last_deactivate_stale_dags_time = timezone.make_aware(datetime.fromtimestamp(0))
        
        self.parsing_cleanup_interval = conf.getint("scheduler", "parsing_cleanup_interval")
        
        self.stale_dag_threshold = conf.getint("scheduler", "stale_dag_threshold")
        
        self._processor_timeout = processor_timeout
        
        self.dag_dir_list_interval = conf.getint("scheduler", "dag_dir_list_interval")
        
        self._callback_to_execute: dict[str, list[CallbackRequest]] = defaultdict(list)
        self._log = logging.getLogger("airflow.processor_manager")
        self.waitables: dict[Any, MultiprocessingConnection | DagFileProcessorProcess] = (
            {
                self._direct_scheduler_conn: self._direct_scheduler_conn,
            }
            if self._direct_scheduler_conn is not None
            else {}
        )
        self.heartbeat: Callable[[], None] = lambda: None
    def register_exit_signals(self):