### 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)

"""Information about single processing of one file."""
    num_dags: int
    import_errors: int
    last_finish_time: datetime | None
    last_duration: timedelta | None
    run_count: int
class DagParsingSignal(enum.Enum):
    """All signals sent to parser."""
    AGENT_RUN_ONCE = "agent_run_once"
    TERMINATE_MANAGER = "terminate_manager"
    END_MANAGER = "end_manager"
class DagFileProcessorAgent(LoggingMixin, MultiprocessingStartMethodMixin):
    """
    Agent for DAG file processing.
    It is responsible for all DAG parsing related jobs in scheduler process.
    Mainly it can spin up DagFileProcessorManager in a subprocess,
    collect DAG parsing results from it and communicate signal/DAG parsing stat with it.
    This class runs in the main `airflow scheduler` process.
    :param dag_directory: Directory where DAG definitions are kept. All
        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 dag_ids: if specified, only schedule tasks with these DAG IDs
    :param pickle_dags: whether to pickle DAGs.
    :param async_mode: Whether to start agent in async mode
    """
    def __init__(