### 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)

"""Start more processors if we have enough slots and files to process."""
        
        
        SecretCache.init()
        while self._parallelism - len(self._processors) > 0 and self._file_path_queue:
            file_path = self._file_path_queue.popleft()
            
            if file_path in self._processors:
                continue
            callback_to_execute_for_file = self._callback_to_execute[file_path]
            processor = self._create_process(
                file_path,
                self._pickle_dags,
                self._dag_ids,
                self.get_dag_directory(),
                callback_to_execute_for_file,
            )
            del self._callback_to_execute[file_path]
            Stats.incr("dag_processing.processes", tags={"file_path": file_path, "action": "start"})
            processor.start()
            self.log.debug("Started a process (PID: %s) to generate tasks for %s", processor.pid, file_path)
            self._processors[file_path] = processor
            self.waitables[processor.waitable_handle] = processor
            Stats.gauge("dag_processing.file_path_queue_size", len(self._file_path_queue))
    def add_new_file_path_to_queue(self):