### 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)

"""Check if the DagFileProcessorManager process is alive, and process any pending messages."""
        if not self._parent_signal_conn:
            raise ValueError("Process not started.")
        
        while self._parent_signal_conn.poll(timeout=0.01):
            try:
                result = self._parent_signal_conn.recv()
            except (EOFError, ConnectionError):
                break
            self._process_message(result)
        
        self._heartbeat_manager()
    def _process_message(self, message):
        self.log.debug("Received message of type %s", type(message).__name__)
        if isinstance(message, DagParsingStat):
            self._sync_metadata(message)
        else:
            raise RuntimeError(f"Unexpected message received of type {type(message).__name__}")
    def _heartbeat_manager(self):