### 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)

"""This is a dag with a setup / teardown at dag level and two task groups that have
        their own setups / teardowns.
        When we do tg >> dag_teardown, teardowns should be excluded from tg leaves.
        """
        dag = DAG(dag_id="test_dag", start_date=pendulum.now())
        with dag:
            dag_setup = BaseOperator(task_id="dag_setup").as_setup()
            dag_teardown = BaseOperator(task_id="dag_teardown").as_teardown()
            dag_setup >> dag_teardown
            for group_name in ("g1", "g2"):
                with TaskGroup(group_name) as tg:
                    group_setup = BaseOperator(task_id="group_setup").as_setup()
                    w1 = BaseOperator(task_id="w1")
                    w2 = BaseOperator(task_id="w2")
                    w3 = BaseOperator(task_id="w3")
                    group_teardown = BaseOperator(task_id="group_teardown").as_teardown()
                    group_setup >> w1 >> w2 >> w3 >> group_teardown
                    group_setup >> group_teardown
                dag_setup >> tg >> dag_teardown
        g2_w2 = dag.task_dict["g2.w2"]
        g2_w3 = dag.task_dict["g2.w3"]
        g2_group_teardown = dag.task_dict["g2.group_teardown"]