### 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)

from airflow.jobs.backfill_job_runner import BackfillJobRunner
        if not executor and local:
            from airflow.executors.local_executor import LocalExecutor
            executor = LocalExecutor()
        elif not executor:
            from airflow.executors.executor_loader import ExecutorLoader
            executor = ExecutorLoader.get_default_executor()
        from airflow.jobs.job import Job
        job = Job(executor=executor)
        job_runner = BackfillJobRunner(
            job=job,
            dag=self,
            start_date=start_date,
            end_date=end_date,
            mark_success=mark_success,
            donot_pickle=donot_pickle,
            ignore_task_deps=ignore_task_deps,
            ignore_first_depends_on_past=ignore_first_depends_on_past,
            pool=pool,
            delay_on_limit_secs=delay_on_limit_secs,
            verbose=verbose,
            conf=conf,
            rerun_failed_tasks=rerun_failed_tasks,
            run_backwards=run_backwards,
            run_at_least_once=run_at_least_once,
            continue_on_failures=continue_on_failures,
            disable_retry=disable_retry,
        )
        run_job(job=job, execute_callable=job_runner._execute)
    def cli(self):