### 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)

if isinstance(child, AbstractOperator):
                    if child.task_id in dag.task_dict:
                        task = copied.children[child.task_id] = dag.task_dict[child.task_id]
                        task.task_group = proxy
                    else:
                        copied.used_group_ids.discard(child.task_id)
                else:
                    filtered_child = filter_task_group(child, proxy)
                    
                    if filtered_child.children:
                        copied.children[child.group_id] = filtered_child
            return copied
        dag._task_group = filter_task_group(self.task_group, None)
        
        
        subdag_task_groups = dag.task_group.get_task_group_dict()
        for group in subdag_task_groups.values():
            group.upstream_group_ids.intersection_update(subdag_task_groups)
            group.downstream_group_ids.intersection_update(subdag_task_groups)
            group.upstream_task_ids.intersection_update(dag.task_dict)
            group.downstream_task_ids.intersection_update(dag.task_dict)
        for t in dag.tasks:
            
            
            t.upstream_task_ids.intersection_update(dag.task_dict)
            t.downstream_task_ids.intersection_update(dag.task_dict)
        if len(dag.tasks) < len(self.tasks):
            dag.partial = True
        return dag
    def has_task(self, task_id: str):