### 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)

for file_path in self.file_paths:
            if file_path not in self._file_stats:
                
                self.log.info("Adding new file %s to parsing queue", file_path)
                self._file_stats[file_path] = DagFileProcessorManager.DEFAULT_FILE_STAT
                self._file_path_queue.appendleft(file_path)
    def prepare_file_path_queue(self):
        """
        Scan dags dir to generate more file paths to process.
        Note this method is only called when the file path queue is empty
        """
        self._parsing_start_time = time.perf_counter()
        
        
        file_paths_in_progress = set(self._processors)
        now = timezone.utcnow()
        
        list_mode = conf.get("scheduler", "file_parsing_sort_mode")
        files_with_mtime = {}
        file_paths = []
        is_mtime_mode = list_mode == "modified_time"
        file_paths_recently_processed = []
        file_paths_to_stop_watching = set()
        for file_path in self._file_paths: