### 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)

else:
                data_interval = dag.get_run_data_interval(run)
            if num_active_runs.get(dag.dag_id, 0) >= orm_dag.max_active_runs:
                orm_dag.next_dagrun_create_after = None
            else:
                orm_dag.calculate_dagrun_date_fields(dag, data_interval)
            dag_tags = set(dag.tags or {})
            orm_dag_tags = list(orm_dag.tags or [])
            for orm_tag in orm_dag_tags:
                if orm_tag.name not in dag_tags:
                    session.delete(orm_tag)
                    orm_dag.tags.remove(orm_tag)
            orm_tag_names = {t.name for t in orm_dag_tags}
            for dag_tag in dag_tags:
                if dag_tag not in orm_tag_names:
                    dag_tag_orm = DagTag(name=dag_tag, dag_id=dag.dag_id)
                    orm_dag.tags.append(dag_tag_orm)
                    session.add(dag_tag_orm)
            orm_dag_links = orm_dag.dag_owner_links or []
            for orm_dag_link in orm_dag_links:
                if orm_dag_link not in dag.owner_links:
                    session.delete(orm_dag_link)
            for owner_name, owner_link in dag.owner_links.items():
                dag_owner_orm = DagOwnerAttributes(dag_id=dag.dag_id, owner=owner_name, link=owner_link)
                session.add(dag_owner_orm)
        DagCode.bulk_sync_to_db(filelocs, session=session)
        from airflow.datasets import Dataset
        from airflow.models.dataset import (
            DagScheduleDatasetReference,
            DatasetModel,