### 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)

external_tasks = session.scalars(query.where(TI.operator == ExternalTaskMarker.__name__))
            for ti in external_tasks:
                ti_key = ti.key.primary
                if ti_key in visited_external_tis:
                    continue
                visited_external_tis.add(ti_key)
                task: ExternalTaskMarker = cast(ExternalTaskMarker, copy.copy(self.get_task(ti.task_id)))
                ti.task = task
                if max_recursion_depth is None:
                    
                    
                    max_recursion_depth = task.recursion_depth
                if recursion_depth + 1 > max_recursion_depth:
                    
                    raise AirflowException(
                        f"Maximum recursion depth {max_recursion_depth} reached for "
                        f"{ExternalTaskMarker.__name__} {ti.task_id}. "
                        f"Attempted to clear too many tasks or there may be a cyclic dependency."
                    )
                ti.render_templates()
                external_tis = session.scalars(
                    select(TI)
                    .join(TI.dag_run)
                    .where(
                        TI.dag_id == task.external_dag_id,
                        TI.task_id == task.external_task_id,
                        DagRun.execution_date == pendulum.parse(task.execution_date),
                    )
                )
                for tii in external_tis: