### 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)

child_pipe, parent_pipe = multiprocessing.Pipe()
        async_mode = "sqlite" not in conf.get("database", "sql_alchemy_conn")
        manager = DagProcessorJobRunner(
            job=Job(),
            processor=DagFileProcessorManager(
                dag_directory=tmpdir,
                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)
        last_runtime = manager.processor.get_last_runtime(manager.processor.file_paths[0])
        child_pipe.close()
        parent_pipe.close()
        statsd_timing_mock.assert_has_calls(
            [
                mock.call("dag_processing.last_duration.temp_dag", timedelta(seconds=last_runtime)),
                mock.call(
                    "dag_processing.last_duration",
                    timedelta(seconds=last_runtime),
                    tags={"file_name": "temp_dag"},
                ),
            ],
            any_order=True,
        )
    def test_refresh_dags_dir_doesnt_delete_zipped_dags(self, tmpdir):