### 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)

local_tz = pendulum.timezone("Europe/Zurich")
        default_args = {"start_date": datetime.datetime(2018, 1, 1, tzinfo=local_tz)}
        dag = DAG("DAG", default_args=default_args)
        assert dag.timezone.name == local_tz.name
        dag = DAG("DAG", default_args=default_args)
        assert dag.timezone.name == local_tz.name
    def test_roots(self):
        """Verify if dag.roots returns the root tasks of a DAG."""
        with DAG("test_dag", start_date=DEFAULT_DATE) as dag:
            op1 = EmptyOperator(task_id="t1")
            op2 = EmptyOperator(task_id="t2")
            op3 = EmptyOperator(task_id="t3")
            op4 = EmptyOperator(task_id="t4")
            op5 = EmptyOperator(task_id="t5")
            [op1, op2] >> op3 >> [op4, op5]
            assert set(dag.roots) == {op1, op2}
    def test_leaves(self):
        """Verify if dag.leaves returns the leaf tasks of a DAG."""
        with DAG("test_dag", start_date=DEFAULT_DATE) as dag:
            op1 = EmptyOperator(task_id="t1")
            op2 = EmptyOperator(task_id="t2")
            op3 = EmptyOperator(task_id="t3")
            op4 = EmptyOperator(task_id="t4")
            op5 = EmptyOperator(task_id="t5")
            [op1, op2] >> op3 >> [op4, op5]
            assert set(dag.leaves) == {op4, op5}
    def test_tree_view(self):