### 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)

manager = DagProcessorJobRunner(
            job=Job(),
            processor=DagFileProcessorManager(
                dag_directory="directory",
                max_runs=1,
                processor_timeout=timedelta(days=365),
                signal_conn=MagicMock(),
                dag_ids=[],
                pickle_dags=False,
                async_mode=True,
            ),
        )
        manager.processor.set_file_paths(dag_files)
        manager.processor.prepare_file_path_queue()
        assert manager.processor._file_path_queue == collections.deque(
            ["file_3.py", "file_2.py", "file_1.py"]
        )
        manager.processor.set_file_paths(dag_files + ["file_4.py"])
        manager.processor.add_new_file_path_to_queue()
        assert manager.processor._file_path_queue == collections.deque(
            ["file_4.py", "file_3.py", "file_2.py", "file_1.py"]
        )
    @conf_vars({("scheduler", "file_parsing_sort_mode"): "modified_time"})
    @mock.patch("airflow.settings.TIMEZONE", timezone.utc)
    @mock.patch("zipfile.is_zipfile", return_value=True)
    @mock.patch("airflow.utils.file.might_contain_dag", return_value=True)
    @mock.patch("airflow.utils.file.find_path_from_directory", return_value=True)
    @mock.patch("airflow.utils.file.os.path.isfile", return_value=True)
    @mock.patch("airflow.utils.file.os.path.getmtime")
    def test_recently_modified_file_is_parsed_with_mtime_mode(