### 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)

child_dag = DAG(
            "parent_dag.child_dag",
            schedule="@daily",
            start_date=DEFAULT_DATE,
        )
        with child_dag:
            EmptyOperator(task_id="a_child")
            EmptyOperator(task_id="b_child")
        parent_dag = DAG(
            "parent_dag",
            schedule="@daily",
            start_date=DEFAULT_DATE,
        )
        
        with parent_dag:
            op1 = EmptyOperator(task_id="a_parent")
            op2 = SubDagOperator(task_id="child_dag", subdag=child_dag)
            op3 = EmptyOperator(task_id="b_parent")
            op1 >> op2 >> op3
        topological_list = parent_dag.topological_sort(include_subdag_tasks=True)
        assert self._occur_before("a_parent", "child_dag", topological_list)
        assert self._occur_before("child_dag", "a_child", topological_list)
        assert self._occur_before("child_dag", "b_child", topological_list)
        assert self._occur_before("a_child", "b_parent", topological_list)
        assert self._occur_before("b_child", "b_parent", topological_list)
    def test_dag_topological_sort_dag_without_tasks(self):
        dag = DAG("dag", start_date=DEFAULT_DATE, default_args={"owner": "owner1"})
        assert () == dag.topological_sort()
    def test_dag_naive_start_date_string(self):