### 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("dag", start_date=DEFAULT_DATE, default_args={"owner": "owner1"}):
            with pytest.raises(AirflowException):
                EmptyOperator(task_id="should_fail", weight_rule="no rule")
    def test_get_num_task_instances(self):
        test_dag_id = "test_get_num_task_instances_dag"
        test_task_id = "task_1"
        test_dag = DAG(dag_id=test_dag_id, start_date=DEFAULT_DATE)
        test_task = EmptyOperator(task_id=test_task_id, dag=test_dag)
        dr1 = test_dag.create_dagrun(state=None, run_id="test1", execution_date=DEFAULT_DATE)
        dr2 = test_dag.create_dagrun(
            state=None, run_id="test2", execution_date=DEFAULT_DATE + datetime.timedelta(days=1)
        )
        dr3 = test_dag.create_dagrun(
            state=None, run_id="test3", execution_date=DEFAULT_DATE + datetime.timedelta(days=2)
        )
        dr4 = test_dag.create_dagrun(
            state=None, run_id="test4", execution_date=DEFAULT_DATE + datetime.timedelta(days=3)
        )
        ti1 = TI(task=test_task, run_id=dr1.run_id)
        ti1.state = None
        ti2 = TI(task=test_task, run_id=dr2.run_id)