### 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)

async_mode=True,
            ),
        )
        manager.processor._run_parsing_loop()
        result = None
        while parent_pipe.poll(timeout=None):
            result = parent_pipe.recv()
            if isinstance(result, DagParsingStat) and result.done:
                break
        
        assert sum(stat.run_count for stat in manager.processor._file_stats.values()) == 3
        with create_session() as session:
            assert session.get(DagModel, dag_id) is not None
    @conf_vars({("core", "load_examples"): "False"})
    @pytest.mark.backend("mysql", "postgres")
    @pytest.mark.execution_timeout(30)
    @mock.patch("airflow.dag_processing.manager.DagFileProcessorProcess")
    def test_pipe_full_deadlock(self, mock_processor):
        dag_filepath = TEST_DAG_FOLDER / "test_scheduler_dags.py"
        child_pipe, parent_pipe = multiprocessing.Pipe()
        
        for fd in (parent_pipe.fileno(),):
            sock = socket.socket(fileno=fd)
            sock.setsockopt(socket.SOL_SOCKET, socket.SO_SNDBUF, 1024)
            sock.setsockopt(socket.SOL_SOCKET, socket.SO_RCVBUF, 1024)
            sock.detach()
        exit_event = threading.Event()
        
        
        def keep_pipe_full(pipe, exit_event):