### 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)

stat = DagFileStat(
                    num_dags=0,
                    import_errors=1,
                    last_finish_time=now,
                    last_duration=duration,
                    run_count=self.get_run_count(file_path) + 1,
                )
                self._file_stats[processor.file_path] = stat
        
        for proc in processors_to_remove:
            self._processors.pop(proc)
    def _add_paths_to_queue(self, file_paths_to_enqueue: list[str], add_at_front: bool):
        """Add stuff to the back or front of the file queue, unless it's already present."""
        new_file_paths = list(p for p in file_paths_to_enqueue if p not in self._file_path_queue)
        if add_at_front:
            self._file_path_queue.extendleft(new_file_paths)
        else:
            self._file_path_queue.extend(new_file_paths)
        Stats.gauge("dag_processing.file_path_queue_size", len(self._file_path_queue))
    def max_runs_reached(self):
        """:return: whether all file paths have been processed max_runs times."""
        if self._max_runs == -1:  
            return False
        for stat in self._file_stats.values():
            if stat.run_count < self._max_runs:
                return False
        if self._num_run < self._max_runs:
            return False
        return True
    def terminate(self):