### 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)

random.Random(get_hostname()).shuffle(file_paths)
        if file_paths_to_stop_watching:
            self.set_file_paths(
                [path for path in self._file_paths if path not in file_paths_to_stop_watching]
            )
        files_paths_at_run_limit = [
            file_path for file_path, stat in self._file_stats.items() if stat.run_count == self._max_runs
        ]
        file_paths_to_exclude = file_paths_in_progress.union(
            file_paths_recently_processed,
            files_paths_at_run_limit,
        )
        
        
        files_paths_to_queue = [
            file_path for file_path in file_paths if file_path not in file_paths_to_exclude
        ]
        for file_path, processor in self._processors.items():
            self.log.debug(
                "File path %s is still being processed (started: %s)",
                processor.file_path,
                processor.start_time.isoformat(),
            )
        self.log.debug("Queuing the following files for processing:\n\t%s", "\n\t".join(files_paths_to_queue))
        for file_path in files_paths_to_queue:
            self._file_stats.setdefault(file_path, DagFileProcessorManager.DEFAULT_FILE_STAT)
        self._add_paths_to_queue(files_paths_to_queue, False)
        Stats.incr("dag_processing.file_path_queue_update_count")
    def _kill_timed_out_processors(self):