### 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)

files in file_paths should be under this directory
    :param max_runs: The number of times to parse and schedule each file. -1
        for unlimited.
    :param processor_timeout: How long to wait before timing out a DAG file processor
    :param signal_conn: connection to communicate signal with processor agent.
    :param dag_ids: if specified, only schedule tasks with these DAG IDs
    :param pickle_dags: whether to pickle DAGs.
    :param async_mode: whether to start the manager in async mode
    """
    DEFAULT_FILE_STAT = DagFileStat(
        num_dags=0, import_errors=0, last_finish_time=None, last_duration=None, run_count=0
    )
    def __init__(
        self,
        dag_directory: os.PathLike[str],
        max_runs: int,
        processor_timeout: timedelta,
        dag_ids: list[str] | None,
        pickle_dags: bool,
        signal_conn: MultiprocessingConnection | None = None,
        async_mode: bool = True,
    ):
        super().__init__()
        
        self._file_paths: list[str] = []
        self._file_path_queue: collections.deque[str] = collections.deque()
        self._max_runs = max_runs
        
        self._direct_scheduler_conn = signal_conn
        self._pickle_dags = pickle_dags