### 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 agent_signal == DagParsingSignal.TERMINATE_MANAGER:
                    self.terminate()
                    break
                elif agent_signal == DagParsingSignal.END_MANAGER:
                    self.end()
                    sys.exit(os.EX_OK)
                elif agent_signal == DagParsingSignal.AGENT_RUN_ONCE:
                    
                    pass
                elif isinstance(agent_signal, CallbackRequest):
                    self._add_callback_to_queue(agent_signal)
                else:
                    raise ValueError(f"Invalid message {type(agent_signal)}")
            if not ready and not self._async_mode:
                
                
                
                
                
                self.log.warning(
                    "wait() unexpectedly returned nothing ready after infinite timeout (%r)!", poll_time
                )
                continue
            for sentinel in ready:
                if sentinel is self._direct_scheduler_conn:
                    continue
                processor = self.waitables.get(sentinel)
                if not processor:
                    continue
                self._collect_results_from_processor(processor)