### 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,
        last_automated_dagrun: None | datetime | DataInterval,
        *,
        restricted: bool = True,
    ) -> DagRunInfo | None:
        """Get information about the next DagRun of this dag after ``date_last_automated_dagrun``.
        This calculates what time interval the next DagRun should operate on
        (its execution date) and when it can be scheduled, according to the
        dag's timetable, start_date, end_date, etc. This doesn't check max
        active run or any other "max_active_tasks" type limits, but only
        performs calculations based on the various date and interval fields of
        this dag and its tasks.
        :param last_automated_dagrun: The ``max(execution_date)`` of
            existing "automated" DagRuns for this dag (scheduled or backfill,
            but not manual).
        :param restricted: If set to *False* (default is *True*), ignore
            ``start_date``, ``end_date``, and ``catchup`` specified on the DAG
            or tasks.
        :return: DagRunInfo of the next dagrun, or None if a dagrun is not
            going to be scheduled.
        """
        
        if self.is_subdag:
            return None