### 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)

"""
        Test that when a processor already exist with a filepath, a new processor won't be created
        with that filepath. The filepath will just be removed from the list.
        """
        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,
            ),
        )
        file_1 = "file_1.py"
        file_2 = "file_2.py"
        file_3 = "file_3.py"
        manager.processor._file_path_queue = collections.deque([file_1, file_2, file_3])
        
        manager.processor._processors[file_1] = MagicMock()
        
        manager.processor.start_new_processes()