### 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)

@dag_decorator(schedule_interval=None)
        def mydag():
            ...
        with pytest.warns(RemovedInAirflow3Warning) as warnings:
            line = sys._getframe().f_lineno + 1
            mydag()
        w = warnings.pop(RemovedInAirflow3Warning)
        assert w.filename == __file__
        assert w.lineno == line
@pytest.mark.parametrize("timetable", [NullTimetable(), OnceTimetable()])
def test_dag_timetable_match_schedule_interval(timetable):
    dag = DAG("my-dag", timetable=timetable)
    assert dag._check_schedule_interval_matches_timetable()
@pytest.mark.parametrize("schedule_interval", [None, "@once", "@daily", timedelta(days=1)])
def test_dag_schedule_interval_match_timetable(schedule_interval):
    dag = DAG("my-dag", schedule=schedule_interval)
    assert dag._check_schedule_interval_matches_timetable()
@pytest.mark.parametrize("schedule_interval", [None, "@daily", timedelta(days=1)])
def test_dag_schedule_interval_change_after_init(schedule_interval):
    dag = DAG("my-dag", timetable=OnceTimetable())
    dag.schedule_interval = schedule_interval
    assert not dag._check_schedule_interval_matches_timetable()
@pytest.mark.parametrize("timetable", [NullTimetable(), OnceTimetable()])
def test_dag_timetable_change_after_init(timetable):