### 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)

"""
        Ensure that datasets referenced in a dag are correctly loaded into the database.
        """
        dag_id1 = "test_dataset_dag1"
        dag_id2 = "test_dataset_dag2"
        task_id = "test_dataset_task"
        uri1 = "s3://dataset1"
        d1 = Dataset(uri1, extra={"not": "used"})
        d2 = Dataset("s3://dataset2")
        d3 = Dataset("s3://dataset3")
        dag1 = DAG(dag_id=dag_id1, start_date=DEFAULT_DATE, schedule=[d1])
        EmptyOperator(task_id=task_id, dag=dag1, outlets=[d2, d3])
        dag2 = DAG(dag_id=dag_id2, start_date=DEFAULT_DATE)
        EmptyOperator(task_id=task_id, dag=dag2, outlets=[Dataset(uri1, extra={"should": "be used"})])
        session = settings.Session()