### 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)

else:
            raise AirflowException(
                f"Invalid values of dag.default_view: only support "
                f"{DEFAULT_VIEW_PRESETS}, but get {default_view}"
            )
        if orientation in ORIENTATION_PRESETS:
            self.orientation = orientation
        else:
            raise AirflowException(
                f"Invalid values of dag.orientation: only support "
                f"{ORIENTATION_PRESETS}, but get {orientation}"
            )
        self.catchup: bool = catchup
        self.partial: bool = False
        self.on_success_callback = on_success_callback
        self.on_failure_callback = on_failure_callback
        
        
        
        self.edge_info: dict[str, dict[str, EdgeInfoType]] = {}
        
        
        self.has_on_success_callback = self.on_success_callback is not None
        self.has_on_failure_callback = self.on_failure_callback is not None
        self._access_control = DAG._upgrade_outdated_dag_access_control(access_control)
        self.is_paused_upon_creation = is_paused_upon_creation
        self.auto_register = auto_register
        self.fail_stop: bool = fail_stop
        self.jinja_environment_kwargs = jinja_environment_kwargs
        self.render_template_as_native_obj = render_template_as_native_obj