### 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)

task_ids: list[str] = []
        if execution_date is None:
            dag_run = session.scalars(
                select(DagRun).where(DagRun.run_id == run_id, DagRun.dag_id == self.dag_id)
            ).one()  
            resolve_execution_date = dag_run.execution_date
        else:
            resolve_execution_date = execution_date
        end_date = resolve_execution_date if not future else None
        start_date = resolve_execution_date if not past else None
        task_group_dict = self.task_group.get_task_group_dict()
        task_group = task_group_dict.get(group_id)
        if task_group is None:
            raise ValueError("TaskGroup {group_id} could not be found")
        tasks_to_set_state = [task for task in task_group.iter_tasks() if isinstance(task, BaseOperator)]
        task_ids = [task.task_id for task in task_group.iter_tasks()]
        dag_runs_query = session.query(DagRun.id).where(DagRun.dag_id == self.dag_id)
        if start_date is None and end_date is None:
            dag_runs_query = dag_runs_query.where(DagRun.execution_date == start_date)
        else:
            if start_date is not None:
                dag_runs_query = dag_runs_query.where(DagRun.execution_date >= start_date)
            if end_date is not None:
                dag_runs_query = dag_runs_query.where(DagRun.execution_date <= end_date)
        with lock_rows(dag_runs_query, session):
            altered = set_state(
                tasks=tasks_to_set_state,
                execution_date=execution_date,
                run_id=run_id,
                upstream=upstream,