### 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 a boolean indicating whether this DAG is paused."""
        return session.scalar(select(DagModel.is_paused).where(DagModel.dag_id == self.dag_id))
    @property
    def is_paused(self):
        """Use `airflow.models.DAG.get_is_paused`, this attribute is deprecated."""
        warnings.warn(
            "This attribute is deprecated. Please use `airflow.models.DAG.get_is_paused` method.",
            RemovedInAirflow3Warning,
            stacklevel=2,
        )
        return self.get_is_paused()
    @property
    def normalized_schedule_interval(self) -> ScheduleInterval:
        warnings.warn(
            "DAG.normalized_schedule_interval() is deprecated.",
            category=RemovedInAirflow3Warning,
            stacklevel=2,
        )
        if isinstance(self.schedule_interval, str) and self.schedule_interval in cron_presets:
            _schedule_interval: ScheduleInterval = cron_presets.get(self.schedule_interval)
        elif self.schedule_interval == "@once":
            _schedule_interval = None
        else:
            _schedule_interval = self.schedule_interval
        return _schedule_interval
    @provide_session
    def handle_callback(self, dagrun, success=True, reason=None, session=NEW_SESSION):