### 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)

dag_ids: list[str], *, session: Session
) -> dict[str, dict[str, int | str]]:
    """
    Get next run info for a list of dag_ids.
    Given a list of dag_ids, get string representing how close any that are dataset triggered are
    their next run, e.g. "1 of 2 datasets updated".
    """
    from airflow.models.dataset import DagScheduleDatasetReference, DatasetDagRunQueue as DDRQ, DatasetModel
    return {
        x.dag_id: {
            "uri": x.uri,
            "ready": x.ready,
            "total": x.total,
        }
        for x in session.execute(
            select(
                DagScheduleDatasetReference.dag_id,
                
                
                case((func.count() == 1, func.max(DatasetModel.uri)), else_="").label("uri"),
                func.count().label("total"),
                func.sum(case((DDRQ.target_dag_id.is_not(None), 1), else_=0)).label("ready"),
            )