### 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)

mock_object(list(my_input))
        with dag:
            mapped_task = get_index.expand(current_val=[1, 1, 1, 1, 1])
            check_task(mapped_task)
        dag.test()
        mock_object.assert_called_with([0, 1, 2, 3, 4])
    def test_dag_connection_file(self):
        test_connections_string = """
---
my_postgres_conn:
  - conn_id: my_postgres_conn
    conn_type: postgres
        """
        dag = DAG(dag_id="test_local_testing_conn_file", start_date=DEFAULT_DATE)
        @task_decorator
        def check_task():
            from airflow.configuration import secrets_backend_list
            from airflow.secrets.local_filesystem import LocalFilesystemBackend
            assert isinstance(secrets_backend_list[0], LocalFilesystemBackend)
            local_secrets: LocalFilesystemBackend = secrets_backend_list[0]
            assert local_secrets.get_connection("my_postgres_conn").conn_id == "my_postgres_conn"
        with dag:
            check_task()
        with NamedTemporaryFile(suffix=".yaml") as tmp:
            with open(tmp.name, "w") as f:
                f.write(test_connections_string)
            dag.test(conn_file_path=tmp.name)
    def _make_test_subdag(self, session):