### 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)

"""
        Deactivate any DAGs that were last touched by the scheduler before the expiration date.
        These DAGs were likely deleted.
        :param expiration_date: set inactive DAGs that were touched before this time
        :return: None
        """
        for dag in session.scalars(
            select(DagModel).where(DagModel.last_parsed_time < expiration_date, DagModel.is_active)
        ):
            log.info(
                "Deactivating DAG ID %s since it was last touched by the scheduler at %s",
                dag.dag_id,
                dag.last_parsed_time.isoformat(),
            )
            dag.is_active = False
            session.merge(dag)
            session.commit()
    @staticmethod
    @provide_session
    def get_num_task_instances(dag_id, run_id=None, task_ids=None, states=None, session=NEW_SESSION) -> int:
        """
        Return the number of task instances in the given DAG.
        :param session: ORM session
        :param dag_id: ID of the DAG to get the task concurrency of