### 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)

session.add(DatasetModel(uri=dataset4.uri, is_orphaned=True))
            session.flush()
            dag1 = DAG(dag_id="datasets-1", start_date=DEFAULT_DATE, schedule=[dataset1])
            BashOperator(dag=dag1, task_id="task", bash_command="echo 1", outlets=[dataset3])
            DAG.bulk_write_to_db([dag1], session=session)
            
            non_orphaned_datasets = [
                dataset.uri
                for dataset in session.query(DatasetModel.uri)
                .filter(~DatasetModel.is_orphaned)
                .order_by(DatasetModel.uri)
            ]
            assert non_orphaned_datasets == ["ds1", "ds3"]
            orphaned_datasets = [
                dataset.uri
                for dataset in session.query(DatasetModel.uri)
                .filter(DatasetModel.is_orphaned)
                .order_by(DatasetModel.uri)
            ]
            assert orphaned_datasets == ["ds2", "ds4"]
            
            dag1 = DAG(dag_id="datasets-1", start_date=DEFAULT_DATE, schedule=[dataset1, dataset2])
            BashOperator(dag=dag1, task_id="task", bash_command="echo 1", outlets=[dataset3, dataset4])
            DAG.bulk_write_to_db([dag1], session=session)
            
            non_orphaned_dataset_count = session.query(DatasetModel).filter(~DatasetModel.is_orphaned).count()
            assert non_orphaned_dataset_count == 4
            orphaned_dataset_count = session.query(DatasetModel).filter(DatasetModel.is_orphaned).count()
            assert orphaned_dataset_count == 0
    def test_sync_to_db(self):