### 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)

.where(DagModel.dag_id.in_(dataset_triggered_dag_ids))
                    .group_by(DagModel.dag_id)
                    .having(func.count() >= func.max(DagModel.max_active_runs))
                )
            )
            if exclusion_list:
                dataset_triggered_dag_ids -= exclusion_list
                dataset_triggered_dag_info = {
                    k: v for k, v in dataset_triggered_dag_info.items() if k not in exclusion_list
                }
        
        query = (
            select(cls)
            .where(
                cls.is_paused == expression.false(),
                cls.is_active == expression.true(),
                cls.has_import_errors == expression.false(),
                or_(
                    cls.next_dagrun_create_after <= func.now(),
                    cls.dag_id.in_(dataset_triggered_dag_ids),
                ),
            )
            .order_by(cls.next_dagrun_create_after)
            .limit(cls.NUM_DAGS_PER_DAGRUN_QUERY)
        )
        return (
            session.scalars(with_row_locks(query, of=cls, session=session, **skip_locked(session=session))),
            dataset_triggered_dag_info,
        )
    def calculate_dagrun_date_fields(