### 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)

return (
            session.query(TI)
            .filter(
                TI.dag_id == dag.dag_id,
                TI.task_id == task.task_id,
                TI.run_id == dagrun.run_id,
            )
            .one()
        )
    get_ti_from_db(task_1).state = State.FAILED
    get_ti_from_db(task_2).state = State.SUCCESS
    get_ti_from_db(task_3).state = State.UPSTREAM_FAILED
    get_ti_from_db(task_4).state = State.SUCCESS
    get_ti_from_db(task_5).state = State.UPSTREAM_FAILED
    get_ti_from_db(task_6).state = State.FAILED
    get_ti_from_db(task_7).state = State.SKIPPED
    session.flush()
    altered = dag.set_task_group_state(
        group_id=section_1.group_id,
        run_id=run_id,
        execution_date=execution_date,
        state=State.SUCCESS,
        session=session,
    )
    
    assert get_ti_from_db(task_1).state == State.SUCCESS