### 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)

"""
    Table defining different owner attributes.
    For example, a link for an owner that will be passed as a hyperlink to the "DAGs" view.
    """
    __tablename__ = "dag_owner_attributes"
    dag_id = Column(
        StringID(),
        ForeignKey("dag.dag_id", name="dag.dag_id", ondelete="CASCADE"),
        nullable=False,
        primary_key=True,
    )
    owner = Column(String(500), primary_key=True, nullable=False)
    link = Column(String(500), nullable=False)
    def __repr__(self):
        return f"<DagOwnerAttributes: dag_id={self.dag_id}, owner={self.owner}, link={self.link}>"
    @classmethod
    def get_all(cls, session) -> dict[str, dict[str, str]]:
        dag_links: dict = collections.defaultdict(dict)
        for obj in session.scalars(select(cls)):
            dag_links[obj.dag_id].update({obj.owner: obj.link})
        return dag_links
class DagModel(Base):