### 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 = DAG(
            "dag",
            start_date=DEFAULT_DATE,
            owner_links={"owner1": "https://mylink.com", "owner2": "mailto:someone@yoursite.com"},
        )
        assert dag.owner_links == {"owner1": "https://mylink.com", "owner2": "mailto:someone@yoursite.com"}
        session = settings.Session()
        dag.sync_to_db(session=session)
        expected_owners = {"dag": {"owner1": "https://mylink.com", "owner2": "mailto:someone@yoursite.com"}}
        orm_dag_owners = DagOwnerAttributes.get_all(session)
        assert orm_dag_owners == expected_owners
        
        dag = DAG(
            "dag",
            start_date=DEFAULT_DATE,
        )
        dag.sync_to_db(session=session)
        orm_dag_owners = session.query(DagOwnerAttributes).all()
        assert not orm_dag_owners
        
        with pytest.raises(AirflowException):