### 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)

"""Heartbeat DAG file processor and restart it if we are not done."""
        if not self._parent_signal_conn:
            raise ValueError("Process not started.")
        if self._process and not self._process.is_alive():
            self._process.join(timeout=0)
            if not self.done:
                self.log.warning(
                    "DagFileProcessorManager (PID=%d) exited with exit code %d - re-launching",
                    self._process.pid,
                    self._process.exitcode,
                )
                self.start()
        if self.done:
            return
        parsing_stat_age = time.monotonic() - self._last_parsing_stat_received_at
        if parsing_stat_age > self._processor_timeout.total_seconds():
            Stats.incr("dag_processing.manager_stalls")
            self.log.error(
                "DagFileProcessorManager (PID=%d) last sent a heartbeat %.2f seconds ago! Restarting it",
                self._process.pid,
                parsing_stat_age,
            )
            reap_process_group(self._process.pid, logger=self.log)
            self.start()
    def _sync_metadata(self, stat):