### 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)

all_files_processed = all(self.get_last_finish_time(x) is not None for x in self.file_paths)
            max_runs_reached = self.max_runs_reached()
            try:
                if self._direct_scheduler_conn:
                    self._direct_scheduler_conn.send(
                        DagParsingStat(
                            max_runs_reached,
                            all_files_processed,
                        )
                    )
            except BlockingIOError:
                
                
                
                
                
                self.log.debug("BlockingIOError received trying to send DagParsingStat, ignoring")
            if max_runs_reached:
                self.log.info(
                    "Exiting dag parsing loop as all files have been processed %s times", self._max_runs
                )
                break
            if self._async_mode:
                loop_duration = time.monotonic() - loop_start_time
                if loop_duration < 1:
                    poll_time = 1 - loop_duration
                else:
                    poll_time = 0.0
    @provide_session
    def _fetch_callbacks(self, max_callbacks: int, session: Session = NEW_SESSION):