### 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)

self.run_processor_manager_one_loop(manager, parent_pipe)
            import_errors = session.query(errors.ImportError).all()
            assert len(import_errors) == 0
            session.rollback()
        child_pipe.close()
        parent_pipe.close()
    @conf_vars({("core", "load_examples"): "False"})
    def test_max_runs_when_no_files(self):
        child_pipe, parent_pipe = multiprocessing.Pipe()
        with TemporaryDirectory(prefix="empty-airflow-dags-") as dags_folder:
            async_mode = "sqlite" not in conf.get("database", "sql_alchemy_conn")
            manager = DagProcessorJobRunner(
                job=Job(),
                processor=DagFileProcessorManager(
                    dag_directory=dags_folder,
                    max_runs=1,
                    processor_timeout=timedelta(days=365),
                    signal_conn=child_pipe,
                    dag_ids=[],
                    pickle_dags=False,
                    async_mode=async_mode,
                ),
            )
            self.run_processor_manager_one_loop(manager, parent_pipe)
        child_pipe.close()
        parent_pipe.close()
    @pytest.mark.backend("mysql", "postgres")
    @mock.patch("airflow.dag_processing.processor.iter_airflow_imports")
    def test_start_new_processes_with_same_filepath(self, _):