### 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)

dataset1 = Dataset(uri="ds1")
        dataset2 = Dataset(uri="ds2")
        for dag_id, dataset in [("datasets-1", dataset1), ("datasets-2", dataset2)]:
            with dag_maker(dag_id=dag_id, start_date=timezone.utcnow(), session=session):
                EmptyOperator(task_id="task", outlets=[dataset])
            dr = dag_maker.create_dagrun()
            ds_id = session.query(DatasetModel.id).filter_by(uri=dataset.uri).scalar()
            session.add(
                DatasetEvent(
                    dataset_id=ds_id,
                    source_task_id="task",
                    source_dag_id=dr.dag_id,
                    source_run_id=dr.run_id,
                    source_map_index=-1,
                )
            )
        ds1_id = session.query(DatasetModel.id).filter_by(uri=dataset1.uri).scalar()
        ds2_id = session.query(DatasetModel.id).filter_by(uri=dataset2.uri).scalar()
        with dag_maker(dag_id="datasets-consumer-multiple", schedule=[dataset1, dataset2]) as dag:
            pass
        session.flush()
        session.add_all(
            [