### 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)

include_dependent_dags=include_subdags,  
            session=session,
            dag_bag=dag_bag,
            exclude_task_ids=exclude_task_ids,
        )
        if dry_run:
            return session.scalars(tis).all()
        tis = session.scalars(tis).all()
        count = len(list(tis))
        do_it = True
        if count == 0:
            return 0
        if confirm_prompt:
            ti_list = "\n".join(str(t) for t in tis)
            question = f"You are about to delete these {count} tasks:\n{ti_list}\n\nAre you sure? [y/n]"
            do_it = utils.helpers.ask_yesno(question)
        if do_it:
            clear_task_instances(
                list(tis),
                session,
                dag=self,
                dag_run_state=dag_run_state,
            )
        else:
            count = 0
            print("Cancelled, nothing was cleared.")
        session.flush()
        return count
    @classmethod
    def clear_dags(