### 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)

"""Helper to return tasks that would be cleared if **upstream** selected."""
        return set(
            task.dag.partial_subset(
                task_ids_or_regex=[task.task_id],
                include_downstream=False,
                include_upstream=False,
            ).tasks
        )
    def test_get_flat_relative_ids_with_setup(self):
        with DAG(dag_id="test_dag", start_date=pendulum.now()) as dag:
            s1, w1, w2, w3, w4, t1 = self.make_tasks(dag, "s1, w1, w2, w3, w4, t1")
        s1 >> w1 >> w2 >> w3
        
        assert set(w1.get_upstreams_only_setups_and_teardowns()) == {s1}
        
        assert set(w2.get_upstreams_only_setups_and_teardowns()) == {s1}
        assert set(w3.get_upstreams_only_setups_and_teardowns()) == {s1}
        
        assert self.cleared_downstream(w2) == {s1, w2, w3}
        w3 >> t1
        
        assert set(w2.get_upstreams_only_setups_and_teardowns()) == {s1}
        
        
        assert self.cleared_downstream(w2) == {s1, w2, w3, t1}