### 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)

"""Print an ASCII tree representation of the DAG."""
        def get_downstream(task, level=0):
            print((" " * level * 4) + str(task))
            level += 1
            for t in task.downstream_list:
                get_downstream(t, level)
        for t in self.roots:
            get_downstream(t)
    @property
    def task(self) -> TaskDecoratorCollection:
        from airflow.decorators import task
        return cast("TaskDecoratorCollection", functools.partial(task, dag=self))
    def add_task(self, task: Operator) -> None:
        """
        Add a task to the DAG.
        :param task: the task you want to add
        """
        FailStopDagInvalidTriggerRule.check(dag=self, trigger_rule=task.trigger_rule)
        from airflow.utils.task_group import TaskGroupContext
        if not self.start_date and not task.start_date:
            raise AirflowException("DAG is missing the start_date parameter")