### 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)

"""
        Update this with a new set of paths to DAG definition files.
        :param new_file_paths: list of paths to DAG definition files
        :return: None
        """
        self._file_paths = new_file_paths
        
        self._file_path_queue = collections.deque(x for x in self._file_path_queue if x in new_file_paths)
        Stats.gauge("dag_processing.file_path_queue_size", len(self._file_path_queue))
        callback_paths_to_del = [x for x in self._callback_to_execute if x not in new_file_paths]
        for path_to_del in callback_paths_to_del:
            del self._callback_to_execute[path_to_del]
        
        filtered_processors = {}
        for file_path, processor in self._processors.items():
            if file_path in new_file_paths:
                filtered_processors[file_path] = processor
            else:
                self.log.warning("Stopping processor for %s", file_path)
                Stats.decr("dag_processing.processes", tags={"file_path": file_path, "action": "stop"})
                processor.terminate()
                self._file_stats.pop(file_path)
        to_remove = set(self._file_stats).difference(self._file_paths)
        for key in to_remove:
            
            del self._file_stats[key]
        self._processors = filtered_processors
    def wait_until_finished(self):