### 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 settings.ALLOW_FUTURE_EXEC_DATES and not self.timetable.can_be_scheduled
    @provide_session
    def get_concurrency_reached(self, session=NEW_SESSION) -> bool:
        """Return a boolean indicating whether the max_active_tasks limit for this DAG has been reached."""
        TI = TaskInstance
        total_tasks = session.scalar(
            select(func.count(TI.task_id)).where(
                TI.dag_id == self.dag_id,
                TI.state == TaskInstanceState.RUNNING,
            )
        )
        return total_tasks >= self.max_active_tasks
    @property
    def concurrency_reached(self):
        """Use `airflow.models.DAG.get_concurrency_reached`, this attribute is deprecated."""
        warnings.warn(
            "This attribute is deprecated. Please use `airflow.models.DAG.get_concurrency_reached` method.",
            RemovedInAirflow3Warning,
            stacklevel=2,
        )
        return self.get_concurrency_reached()
    @provide_session
    def get_is_active(self, session=NEW_SESSION) -> None:
        """Return a boolean indicating whether this DAG is active."""
        return session.scalar(select(DagModel.is_active).where(DagModel.dag_id == self.dag_id))
    @provide_session
    def get_is_paused(self, session=NEW_SESSION) -> None: