### 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)

user_defined_filters: dict | None = None,
    default_args: dict | None = None,
    concurrency: int | None = None,
    max_active_tasks: int = airflow_conf.getint("core", "max_active_tasks_per_dag"),
    max_active_runs: int = airflow_conf.getint("core", "max_active_runs_per_dag"),
    dagrun_timeout: timedelta | None = None,
    sla_miss_callback: None | SLAMissCallback | list[SLAMissCallback] = None,
    default_view: str = airflow_conf.get_mandatory_value("webserver", "dag_default_view").lower(),
    orientation: str = airflow_conf.get_mandatory_value("webserver", "dag_orientation"),
    catchup: bool = airflow_conf.getboolean("scheduler", "catchup_by_default"),
    on_success_callback: None | DagStateChangeCallback | list[DagStateChangeCallback] = None,
    on_failure_callback: None | DagStateChangeCallback | list[DagStateChangeCallback] = None,
    doc_md: str | None = None,
    params: collections.abc.MutableMapping | None = None,
    access_control: dict | None = None,
    is_paused_upon_creation: bool | None = None,
    jinja_environment_kwargs: dict | None = None,
    render_template_as_native_obj: bool = False,
    tags: list[str] | None = None,
    owner_links: dict[str, str] | None = None,
    auto_register: bool = True,
    fail_stop: bool = False,
) -> Callable[[Callable], Callable[..., DAG]]:
    """
    Python dag decorator which wraps a function into an Airflow DAG.
    Accepts kwargs for operator kwarg. Can be used to parameterize DAGs.
    :param dag_args: Arguments for DAG object
    :param dag_kwargs: Kwargs for DAG object.
    """
    def wrapper(f: Callable) -> Callable[..., DAG]: