### 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)

if min_update_interval is not None:
            if session.scalar(
                select(literal(True)).where(
                    cls.dag_id == dag.dag_id,
                    (timezone.utcnow() - timedelta(seconds=min_update_interval)) < cls.last_updated,
                )
            ):
                return False
        log.debug("Checking if DAG (%s) changed", dag.dag_id)
        new_serialized_dag = cls(dag, processor_subdir)
        serialized_dag_db = session.execute(
            select(cls.dag_hash, cls.processor_subdir).where(cls.dag_id == dag.dag_id)
        ).first()
        if (
            serialized_dag_db is not None
            and serialized_dag_db.dag_hash == new_serialized_dag.dag_hash
            and serialized_dag_db.processor_subdir == new_serialized_dag.processor_subdir
        ):
            log.debug("Serialized DAG (%s) is unchanged. Skipping writing to DB", dag.dag_id)
            return False
        log.debug("Writing Serialized DAG: %s to the DB", dag.dag_id)
        session.merge(new_serialized_dag)
        log.debug("DAG: %s written to the DB", dag.dag_id)
        return True
    @classmethod
    @provide_session
    def read_all_dags(cls, session: Session = NEW_SESSION) -> dict[str, SerializedDAG]: