### 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)

.order_by(TI.run_id, TI.task_id, TI.map_index)
    )
    
    assert ti_query.all() == [
        ("downstream", -1, dr1.run_id, TaskInstanceState.FAILED),
        (task_id, 0, dr1.run_id, TaskInstanceState.FAILED),
        (task_id, 1, dr1.run_id, TaskInstanceState.FAILED),
        ("downstream", -1, dr2.run_id, TaskInstanceState.FAILED),
        (task_id, 0, dr2.run_id, TaskInstanceState.FAILED),
        (task_id, 1, dr2.run_id, TaskInstanceState.FAILED),
    ]
    dag.set_task_instance_state(
        task_id=task_id,
        map_indexes=[1],
        future=True,
        run_id=dr1.run_id,
        state=TaskInstanceState.SUCCESS,
        session=session,
    )
    assert dr1 in session, "Check session is passed down all the way"
    assert ti_query.all() == [
        ("downstream", -1, dr1.run_id, None),
        (task_id, 0, dr1.run_id, TaskInstanceState.FAILED),
        (task_id, 1, dr1.run_id, TaskInstanceState.SUCCESS),
        ("downstream", -1, dr2.run_id, None),
        (task_id, 0, dr2.run_id, TaskInstanceState.FAILED),
        (task_id, 1, dr2.run_id, TaskInstanceState.SUCCESS),
    ]
@pytest.mark.parametrize("run_id, execution_date", [(None, datetime_tz(2020, 1, 1)), ("test-run-id", None)])
def test_set_task_group_state(run_id, execution_date, session, dag_maker):