### 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)

"""
        Make sure DST transitions are properly observed
        """
        local_tz = pendulum.timezone("Europe/Zurich")
        start = local_tz.convert(datetime.datetime(2018, 10, 28, 2, 55), dst_rule=pendulum.PRE_TRANSITION)
        assert start.isoformat() == "2018-10-28T02:55:00+02:00", "Pre-condition: start date is in DST"
        utc = timezone.convert_to_utc(start)
        assert utc.isoformat() == "2018-10-28T00:55:00+00:00", "Pre-condition: correct DST->UTC conversion"
        dag = DAG("tz_dag", start_date=start, schedule="*/5 * * * *")
        _next = dag.following_schedule(utc)
        next_local = local_tz.convert(_next)
        assert _next.isoformat() == "2018-10-28T01:00:00+00:00"
        assert next_local.isoformat() == "2018-10-28T02:00:00+01:00"
        prev = dag.previous_schedule(utc)
        prev_local = local_tz.convert(prev)
        assert prev_local.isoformat() == "2018-10-28T02:50:00+02:00"
        prev = dag.previous_schedule(_next)
        prev_local = local_tz.convert(prev)
        assert prev_local.isoformat() == "2018-10-28T02:55:00+02:00"
        assert prev == utc
    def test_following_previous_schedule_daily_dag_cest_to_cet(self):