### 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)

else:
            self.log.debug("Queuing %s CallbackRequest: %s", type(request).__name__, request)
            self._callback_to_execute[request.full_filepath].append(request)
            if request.full_filepath in self._file_path_queue:
                
                
                
                self._file_path_queue = collections.deque(
                    file_path for file_path in self._file_path_queue if file_path != request.full_filepath
                )
            self._add_paths_to_queue([request.full_filepath], True)
            Stats.incr("dag_processing.other_callback_count")
    def _refresh_dag_dir(self) -> bool:
        """Refresh file paths from dag dir if we haven't done it for too long."""
        now = timezone.utcnow()
        elapsed_time_since_refresh = (now - self.last_dag_dir_refresh_time).total_seconds()
        if elapsed_time_since_refresh > self.dag_dir_list_interval:
            
            self.log.info("Searching for files in %s", self._dag_directory)
            self._file_paths = list_py_file_paths(self._dag_directory)
            self.last_dag_dir_refresh_time = now
            self.log.info("There are %s files in %s", len(self._file_paths), self._dag_directory)
            self.set_file_paths(self._file_paths)
            try:
                self.log.debug("Removing old import errors")
                DagFileProcessorManager.clear_nonexistent_import_errors(file_paths=self._file_paths)
            except Exception:
                self.log.exception("Error removing old import errors")
            def _iter_dag_filelocs(fileloc: str) -> Iterator[str]: