### 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)

self.waitables.pop(sentinel)
                self._processors.pop(processor.file_path)
            if self.standalone_dag_processor:
                self._fetch_callbacks(max_callbacks_per_loop)
            self._scan_stale_dags()
            DagWarning.purge_inactive_dag_warnings()
            refreshed_dag_dir = self._refresh_dag_dir()
            self._kill_timed_out_processors()
            
            
            
            if not self._file_path_queue:
                self.emit_metrics()
                self.prepare_file_path_queue()
            
            elif refreshed_dag_dir:
                self.add_new_file_path_to_queue()
            self.start_new_processes()
            
            self._num_run += 1
            if not self._async_mode:
                self.log.debug("Waiting for processors to finish since we're using sqlite")
                
                
                
                
                self.wait_until_finished()
            
            self.collect_results()
            self._print_stat()