### 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)

assert "dag_teardown" not in g2_group_teardown.downstream_task_ids
        
        assert g2_group_teardown.downstream_task_ids == set()
        
        
        assert g2_w3.downstream_task_ids == {"g2.group_teardown", "dag_teardown"}
        
        assert dag_teardown.upstream_task_ids == {"g1.w3", "g2.w3", "dag_setup"}
        assert {x.task_id for x in g2_w2.get_upstreams_only_setups_and_teardowns()} == {
            "dag_setup",
            "dag_teardown",
            "g2.group_setup",
            "g2.group_teardown",
        }
        
        
        assert {x.task_id for x in self.cleared_downstream(g2_w2)} == {
            "dag_setup",
            "dag_teardown",
            "g2.group_setup",
            "g2.group_teardown",
            "g2.w3",
            "g2.w2",
        }
        assert {x.task_id for x in self.cleared_upstream(g2_w2)} == {
            "dag_setup",
            "dag_teardown",
            "g2.group_setup",
            "g2.group_teardown",