### 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)

self,
        execution_date: datetime | None = None,
        run_id: str | None = None,
        session: Session = NEW_SESSION,
    ):
        """
        Return the dag run for a given execution date or run_id if it exists, otherwise none.
        :param execution_date: The execution date of the DagRun to find.
        :param run_id: The run_id of the DagRun to find.
        :param session:
        :return: The DagRun if found, otherwise None.
        """
        if not (execution_date or run_id):
            raise TypeError("You must provide either the execution_date or the run_id")
        query = select(DagRun)
        if execution_date:
            query = query.where(DagRun.dag_id == self.dag_id, DagRun.execution_date == execution_date)
        if run_id:
            query = query.where(DagRun.dag_id == self.dag_id, DagRun.run_id == run_id)
        return session.scalar(query)
    @provide_session
    def get_dagruns_between(self, start_date, end_date, session=NEW_SESSION):