### 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)

"""Kill any file processors that timeout to defend against process hangs."""
        now = timezone.utcnow()
        processors_to_remove = []
        for file_path, processor in self._processors.items():
            duration = now - processor.start_time
            if duration > self._processor_timeout:
                self.log.error(
                    "Processor for %s with PID %s started at %s has timed out, killing it.",
                    file_path,
                    processor.pid,
                    processor.start_time.isoformat(),
                )
                Stats.decr("dag_processing.processes", tags={"file_path": file_path, "action": "timeout"})
                Stats.incr("dag_processing.processor_timeouts", tags={"file_path": file_path})
                
                Stats.incr("dag_file_processor_timeouts")
                processor.kill()
                
                self.waitables.pop(processor.waitable_handle)
                processors_to_remove.append(file_path)