### 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)

dags: list[DAG],
        processor_subdir: str | None = None,
        session: Session = NEW_SESSION,
    ) -> None:
        """
        Save DAGs as Serialized DAG objects in the database.
        Each DAG is saved in a separate database query.
        :param dags: the DAG objects to save to the DB
        :param session: ORM Session
        :return: None
        """
        for dag in dags:
            if not dag.is_subdag:
                SerializedDagModel.write_dag(
                    dag=dag,
                    min_update_interval=MIN_SERIALIZED_DAG_UPDATE_INTERVAL,
                    processor_subdir=processor_subdir,
                    session=session,
                )
    @classmethod
    @provide_session
    def get_last_updated_datetime(cls, dag_id: str, session: Session = NEW_SESSION) -> datetime | None: