### 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)

session.flush()
        subdag = self.partial_subset(
            task_ids_or_regex={task_id},
            include_downstream=True,
            include_upstream=False,
        )
        if execution_date is None:
            dag_run = session.scalars(
                select(DagRun).where(DagRun.run_id == run_id, DagRun.dag_id == self.dag_id)
            ).one()  
            resolve_execution_date = dag_run.execution_date
        else:
            resolve_execution_date = execution_date
        end_date = resolve_execution_date if not future else None
        start_date = resolve_execution_date if not past else None
        subdag.clear(
            start_date=start_date,
            end_date=end_date,
            include_subdags=True,
            include_parentdag=True,
            only_failed=True,
            session=session,
            
            exclude_task_ids=frozenset({task_id}),
        )
        return altered
    @provide_session
    def set_task_group_state(