### 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)

if last_run:
                seconds_ago = (now - last_run).total_seconds()
                Stats.gauge(f"dag_processing.last_run.seconds_ago.{file_name}", seconds_ago)
            rows.append((file_path, processor_pid, runtime, num_dags, num_errors, last_runtime, last_run))
        
        rows = sorted(rows, key=lambda x: x[3] or 0.0)
        formatted_rows = []
        for file_path, pid, runtime, num_dags, num_errors, last_runtime, last_run in rows:
            formatted_rows.append(
                (
                    file_path,
                    pid,
                    f"{runtime.total_seconds():.2f}s" if runtime else None,
                    num_dags,
                    num_errors,
                    f"{last_runtime:.2f}s" if last_runtime else None,
                    last_run.strftime("%Y-%m-%dT%H:%M:%S") if last_run else None,
                )
            )
        log_str = (
            "\n"
            + "=" * 80
            + "\n"
            + "DAG File Processing Stats\n\n"
            + tabulate(formatted_rows, headers=headers)
            + "\n"
            + "=" * 80
        )
        self.log.info(log_str)
    def get_pid(self, file_path) -> int | None: