### 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("dag", start_date=DEFAULT_DATE, owner_links={"owner1": "my-bad-link"})
    @pytest.mark.parametrize(
        "kwargs",
        [
            {"schedule_interval": "@daily", "schedule": "@weekly"},
            {"timetable": NullTimetable(), "schedule": "@weekly"},
            {"timetable": NullTimetable(), "schedule_interval": "@daily"},
        ],
        ids=[
            "schedule_interval+schedule",
            "timetable+schedule",
            "timetable+schedule_interval",
        ],
    )
    def test_schedule_dag_param(self, kwargs):
        with pytest.raises(ValueError, match="At most one"):
            with DAG(dag_id="hello", **kwargs):
                pass
    def test_continuous_schedule_interval_limits_max_active_runs(self):
        dag = DAG("continuous", start_date=DEFAULT_DATE, schedule_interval="@continuous", max_active_runs=1)
        assert isinstance(dag.timetable, ContinuousTimetable)
        assert dag.max_active_runs == 1
        dag = DAG("continuous", start_date=DEFAULT_DATE, schedule_interval="@continuous", max_active_runs=0)
        assert isinstance(dag.timetable, ContinuousTimetable)
        assert dag.max_active_runs == 0
        with pytest.raises(AirflowException):
            dag = DAG(
                "continuous", start_date=DEFAULT_DATE, schedule_interval="@continuous", max_active_runs=25
            )
class TestDagModel: