### 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)

"""
        Get the latest DAG version for a given DAG ID.
        :param dag_id: DAG ID
        :param session: ORM Session
        :return: DAG Hash, or None if the DAG is not found
        """
        return session.scalar(select(cls.dag_hash).where(cls.dag_id == dag_id))
    @classmethod
    def get_latest_version_hash_and_updated_datetime(
        cls,
        dag_id: str,
        *,
        session: Session,
    ) -> tuple[str, datetime] | None:
        """
        Get the latest version for a DAG ID and the date it was last updated in serialized_dag table.
        :meta private:
        :param dag_id: DAG ID
        :param session: ORM Session
        :return: A tuple of DAG Hash and last updated datetime, or None if the DAG is not found
        """
        return session.execute(
            select(cls.dag_hash, cls.last_updated).where(cls.dag_id == dag_id)
        ).one_or_none()
    @classmethod
    @provide_session
    def get_dag_dependencies(cls, session: Session = NEW_SESSION) -> dict[str, list[DagDependency]]: