### 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)

return sorted(x.task_id for x in task_list)
        
        
        assert self.cleared_downstream(w1) == {s1, w1, t1, t2}
        
        
        assert self.cleared_neither(w1) == {s1, w1, t1}
        assert self.cleared_upstream(w1) == {s1, w1, t1}
        
        
        
        assert self.cleared_downstream(t1) == {t1, t2}
        
        assert self.cleared_neither(t1) == {t1}
        
        assert self.cleared_upstream(t1) == {s1, t1, s2, t2, w1}
    def test_clearing_behavior_just_teardown(self):
        with DAG(dag_id="test_dag", start_date=pendulum.now()) as dag:
            s1, t1 = self.make_tasks(dag, "s1, t1")
        s1 >> t1
        assert set(t1.get_upstreams_only_setups_and_teardowns()) == set()
        assert self.cleared_upstream(t1) == {s1, t1}
        assert self.cleared_downstream(t1) == {t1}
        assert self.cleared_neither(t1) == {t1}
        assert set(s1.get_upstreams_only_setups_and_teardowns()) == set()
        assert self.cleared_upstream(s1) == {s1, t1}
        assert self.cleared_downstream(s1) == {s1, t1}
        assert self.cleared_neither(s1) == {s1, t1}