### 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)

cls,
        last_parsed: dict[str, datetime | None],
        dag_directory: str,
        stale_dag_threshold: int,
        session: Session = NEW_SESSION,
    ):
        """
        Detect DAGs which are no longer present in files.
        Deactivate them and remove them in the serialized_dag table.
        """
        to_deactivate = set()
        query = session.query(DagModel.dag_id, DagModel.fileloc, DagModel.last_parsed_time).filter(
            DagModel.is_active
        )
        standalone_dag_processor = conf.getboolean("scheduler", "standalone_dag_processor")
        if standalone_dag_processor:
            query = query.filter(DagModel.processor_subdir == dag_directory)
        dags_parsed = query.all()
        for dag in dags_parsed:
            
            
            
            
            if (
                dag.fileloc in last_parsed
                and (dag.last_parsed_time + timedelta(seconds=stale_dag_threshold)) < last_parsed[dag.fileloc]
            ):
                cls.logger().info("DAG %s is missing and will be deactivated.", dag.dag_id)
                to_deactivate.add(dag.dag_id)
        if to_deactivate: