### 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)

assert op1.dag is dag
        assert op1.owner == "owner1"
        assert op2.dag is dag2
        assert op2.owner == "owner2"
        with dag2:
            op3 = EmptyOperator(task_id="op3")
        assert op3.dag is dag2
        assert op3.owner == "owner2"
        with dag:
            with dag2:
                op4 = EmptyOperator(task_id="op4")
            op5 = EmptyOperator(task_id="op5")
        assert op4.dag is dag2
        assert op5.dag is dag
        assert op4.owner == "owner2"
        assert op5.owner == "owner1"
        with DAG("creating_dag_in_cm", start_date=DEFAULT_DATE) as dag:
            EmptyOperator(task_id="op6")
        assert dag.dag_id == "creating_dag_in_cm"
        assert dag.tasks[0].task_id == "op6"
        with dag:
            with dag:
                op7 = EmptyOperator(task_id="op7")
            op8 = EmptyOperator(task_id="op8")
        op9 = EmptyOperator(task_id="op8")
        op9.dag = dag2
        assert op7.dag == dag
        assert op8.dag == dag
        assert op9.dag == dag2
    def test_dag_topological_sort_include_subdag_tasks(self):