### 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)

catchup=catchup,
                on_success_callback=on_success_callback,
                on_failure_callback=on_failure_callback,
                doc_md=doc_md,
                params=params,
                access_control=access_control,
                is_paused_upon_creation=is_paused_upon_creation,
                jinja_environment_kwargs=jinja_environment_kwargs,
                render_template_as_native_obj=render_template_as_native_obj,
                tags=tags,
                schedule=schedule,
                owner_links=owner_links,
                auto_register=auto_register,
                fail_stop=fail_stop,
            ) as dag_obj:
                
                if f.__doc__ and not dag_obj.doc_md:
                    dag_obj.doc_md = f.__doc__
                
                
                f_kwargs = {}
                for name, value in f_sig.arguments.items():
                    f_kwargs[name] = dag_obj.param(name, value)
                
                back = sys._getframe().f_back
                dag_obj.fileloc = back.f_code.co_filename if back else ""
                
                f(**f_kwargs)
            
            return dag_obj