### 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)

if is_mtime_mode:
                try:
                    files_with_mtime[file_path] = os.path.getmtime(file_path)
                except FileNotFoundError:
                    self.log.warning("Skipping processing of missing file: %s", file_path)
                    self._file_stats.pop(file_path, None)
                    file_paths_to_stop_watching.add(file_path)
                    continue
                file_modified_time = datetime.fromtimestamp(files_with_mtime[file_path], tz=timezone.utc)
            else:
                file_paths.append(file_path)
                file_modified_time = None
            
            
            
            
            last_finish_time = self.get_last_finish_time(file_path)
            if (
                last_finish_time is not None
                and (now - last_finish_time).total_seconds() < self._file_process_interval
                and not (is_mtime_mode and file_modified_time and (file_modified_time > last_finish_time))
            ):
                file_paths_recently_processed.append(file_path)
        
        if is_mtime_mode:
            file_paths = sorted(files_with_mtime, key=files_with_mtime.get, reverse=True)
        elif list_mode == "alphabetical":
            file_paths = sorted(file_paths)
        elif list_mode == "random_seeded_by_host":