### 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)

"""Stop all running processors."""
        for processor in self._processors.values():
            Stats.decr(
                "dag_processing.processes", tags={"file_path": processor.file_path, "action": "terminate"}
            )
            processor.terminate()
    def end(self):
        """Kill all child processes on exit since we don't want to leave them as orphaned."""
        pids_to_kill = self.get_all_pids()
        if pids_to_kill:
            kill_child_processes_by_pids(pids_to_kill)
    def emit_metrics(self):
        """
        Emit metrics about dag parsing summary.
        This is called once every time around the parsing "loop" - i.e. after
        all files have been parsed.
        """
        parse_time = time.perf_counter() - self._parsing_start_time
        Stats.gauge("dag_processing.total_parse_time", parse_time)
        Stats.gauge("dagbag_size", sum(stat.num_dags for stat in self._file_stats.values()))
        Stats.gauge(
            "dag_processing.import_errors", sum(stat.import_errors for stat in self._file_stats.values())
        )
    @property
    def file_paths(self):
        return self._file_paths