### 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)

for node in group.topological_sort(_include_subdag_tasks=include_subdag_tasks):
                if isinstance(node, TaskGroup):
                    yield from nested_topo(node)
                else:
                    yield node
        return tuple(nested_topo(self.task_group))
    @provide_session
    def set_dag_runs_state(
        self,
        state: DagRunState = DagRunState.RUNNING,
        session: Session = NEW_SESSION,
        start_date: datetime | None = None,
        end_date: datetime | None = None,
        dag_ids: list[str] = [],
    ) -> None:
        warnings.warn(
            "This method is deprecated and will be removed in a future version.",
            RemovedInAirflow3Warning,
            stacklevel=3,
        )
        dag_ids = dag_ids or [self.dag_id]
        query = update(DagRun).where(DagRun.dag_id.in_(dag_ids))
        if start_date:
            query = query.where(DagRun.execution_date >= start_date)
        if end_date:
            query = query.where(DagRun.execution_date <= end_date)
        session.execute(query.values(state=state).execution_options(synchronize_session="fetch"))
    @provide_session
    def clear(