### 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)

processor_timeout=timedelta(seconds=5),
            signal_conn=child_pipe,
            pickle_dags=False,
            async_mode=True,
        )
        try:
            thread.start()
            
            manager._run_parsing_loop()
            exit_event.set()
        finally:
            logging.info("Closing pipes")
            parent_pipe.close()
            child_pipe.close()
            logging.info("Closed pipes")
            logging.info("Joining thread")
            thread.join(timeout=1.0)
            logging.info("Joined thread")
    @conf_vars({("core", "load_examples"): "False"})
    @mock.patch("airflow.dag_processing.manager.Stats.timing")
    def test_send_file_processing_statsd_timing(self, statsd_timing_mock, tmpdir):
        filename_to_parse = tmpdir / "temp_dag.py"
        dag_code = dedent(
            """
        from airflow import DAG
        dag = DAG(dag_id='temp_dag', schedule='0 0 * * *')
        """
        )
        with open(filename_to_parse, "w") as file_to_parse:
            file_to_parse.writelines(dag_code)