### 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)

return f"<DAG: {self.dag_id}>"
    @property
    def next_dagrun_data_interval(self) -> DataInterval | None:
        return _get_model_data_interval(
            self,
            "next_dagrun_data_interval_start",
            "next_dagrun_data_interval_end",
        )
    @next_dagrun_data_interval.setter
    def next_dagrun_data_interval(self, value: tuple[datetime, datetime] | None) -> None:
        if value is None:
            self.next_dagrun_data_interval_start = self.next_dagrun_data_interval_end = None
        else:
            self.next_dagrun_data_interval_start, self.next_dagrun_data_interval_end = value
    @property
    def timezone(self):
        return settings.TIMEZONE
    @staticmethod
    @provide_session
    def get_dagmodel(dag_id: str, session: Session = NEW_SESSION) -> DagModel | None:
        return session.get(
            DagModel,
            dag_id,
            options=[joinedload(DagModel.parent_dag)],
        )
    @classmethod
    @provide_session
    def get_current(cls, dag_id, session=NEW_SESSION):