### 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)

def next_dagrun_info(self, last_automated_data_interval, restriction):
                raise RuntimeError("this fails")
        dag = DAG(
            "test_next_dagrun_info_timetable_exception",
            start_date=timezone.datetime(2020, 5, 1),
            timetable=FailingTimetable(),
            catchup=True,
        )
        def _check_logs(records: list[logging.LogRecord], data_interval: DataInterval) -> None:
            assert len(records) == 1
            record = records[0]
            assert record.exc_info is not None, "Should contain exception"
            assert record.getMessage() == (
                f"Failed to fetch run info after data interval {data_interval} "
                f"for DAG 'test_next_dagrun_info_timetable_exception'"
            )
        with caplog.at_level(level=logging.ERROR):
            next_info = dag.next_dagrun_info(None)
        assert next_info is None, "failed next_dagrun_info should return None"
        _check_logs(caplog.records, data_interval=None)
        caplog.clear()
        data_interval = DataInterval(timezone.datetime(2020, 5, 1), timezone.datetime(2020, 5, 2))
        with caplog.at_level(level=logging.ERROR):
            next_info = dag.next_dagrun_info(data_interval)
        assert next_info is None, "failed next_dagrun_info should return None"
        _check_logs(caplog.records, data_interval)
    def test_next_dagrun_after_auto_align(self):