### 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)

if should_pass:
        models.DAG("test-dag", tags=tags)
    else:
        with pytest.raises(AirflowException):
            models.DAG("test-dag", tags=tags)
@pytest.mark.need_serialized_dag
def test_get_dataset_triggered_next_run_info(dag_maker, clear_datasets):
    dataset1 = Dataset(uri="ds1")
    dataset2 = Dataset(uri="ds2")
    dataset3 = Dataset(uri="ds3")
    with dag_maker(dag_id="datasets-1", schedule=[dataset2]):
        pass
    dag1 = dag_maker.dag
    with dag_maker(dag_id="datasets-2", schedule=[dataset1, dataset2]):
        pass
    dag2 = dag_maker.dag
    with dag_maker(dag_id="datasets-3", schedule=[dataset1, dataset2, dataset3]):
        pass
    dag3 = dag_maker.dag