### 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)

DataInterval(DEFAULT_DATE, DEFAULT_DATE + timedelta(days=2)),
            id="modern",
        ),
        pytest.param(
            DEFAULT_DATE,
            None,
            None,
            DataInterval(DEFAULT_DATE, DEFAULT_DATE + timedelta(days=1)),
            id="legacy",
        ),
    ],
)
def test_get_next_data_interval(
    logical_date,
    data_interval_start,
    data_interval_end,
    expected_data_interval,
):
    dag = DAG(dag_id="test_get_next_data_interval", schedule="@daily")
    dag_model = DagModel(
        dag_id="test_get_next_data_interval",
        next_dagrun=logical_date,
        next_dagrun_data_interval_start=data_interval_start,
        next_dagrun_data_interval_end=data_interval_end,
    )
    assert dag.get_next_data_interval(dag_model) == expected_data_interval
@pytest.mark.parametrize(
    ("dag_date", "tasks_date", "restrict"),
    [
        [