### 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)

return session.scalar(select(cls).where(cls.dag_id == dag_id))
    @provide_session
    def get_last_dagrun(self, session=NEW_SESSION, include_externally_triggered=False):
        return get_last_dagrun(
            self.dag_id, session=session, include_externally_triggered=include_externally_triggered
        )
    def get_is_paused(self, *, session: Session | None = None) -> bool:
        """Provide interface compatibility to 'DAG'."""
        return self.is_paused
    @staticmethod
    @internal_api_call
    @provide_session
    def get_paused_dag_ids(dag_ids: list[str], session: Session = NEW_SESSION) -> set[str]:
        """
        Given a list of dag_ids, get a set of Paused Dag Ids.
        :param dag_ids: List of Dag ids
        :param session: ORM Session
        :return: Paused Dag_ids
        """
        paused_dag_ids = session.execute(
            select(DagModel.dag_id)
            .where(DagModel.is_paused == expression.true())
            .where(DagModel.dag_id.in_(dag_ids))
        )
        paused_dag_ids = {paused_dag_id for paused_dag_id, in paused_dag_ids}
        return paused_dag_ids
    def get_default_view(self) -> str: