### 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)

"template_searchpath",
        "last_loaded",
    }
    __serialized_fields: frozenset[str] | None = None
    fileloc: str
    """
    File path that needs to be imported to load this DAG or subdag.
    This may not be an actual file on disk in the case when this DAG is loaded
    from a ZIP file or other DAG distribution format.
    """
    parent_dag: DAG | None = None  
    
    
    def __init__(
        self,
        dag_id: str,
        description: str | None = None,
        schedule: ScheduleArg = NOTSET,
        schedule_interval: ScheduleIntervalArg = NOTSET,
        timetable: Timetable | None = None,
        start_date: datetime | None = None,
        end_date: datetime | None = None,
        full_filepath: str | None = None,
        template_searchpath: str | Iterable[str] | None = None,
        template_undefined: type[jinja2.StrictUndefined] = jinja2.StrictUndefined,
        user_defined_macros: dict | None = None,
        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"),