### 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)

with DAG("test_dag", start_date=DEFAULT_DATE) as dag:
            start = EmptyOperator(task_id="start")
            with TaskGroup(group_id="outer", prefix_group_id=False) as outer_group:
                with TaskGroup(group_id="tg1", prefix_group_id=False) as tg1:
                    EmptyOperator(task_id="t1")
                with TaskGroup(group_id="tg2", prefix_group_id=False) as tg2:
                    EmptyOperator(task_id="t2")
                start >> tg1 >> tg2
        
        task = dag.get_task("t2")
        assert task.task_group.upstream_group_ids == {"tg1"}
        assert isinstance(task.task_group.parent_group, weakref.ProxyType)
        assert task.task_group.parent_group == outer_group
        partial = dag.partial_subset(["t2"], include_upstream=True, include_downstream=False)
        copied_task = partial.get_task("t2")
        assert copied_task.task_group.upstream_group_ids == {"tg1"}
        assert isinstance(copied_task.task_group.parent_group, weakref.ProxyType)
        assert copied_task.task_group.parent_group
        
        assert task.task_group.upstream_group_ids is not copied_task.task_group.upstream_group_ids
    def test_schedule_dag_no_previous_runs(self):