### 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)

"""Test that set_task_group_state updates the TaskGroup state and clear downstream failed"""
    start_date = datetime_tz(2020, 1, 1)
    with dag_maker("test_set_task_group_state", start_date=start_date, session=session) as dag:
        start = EmptyOperator(task_id="start")
        with TaskGroup("section_1", tooltip="Tasks for section_1") as section_1:
            task_1 = EmptyOperator(task_id="task_1")
            task_2 = EmptyOperator(task_id="task_2")
            task_3 = EmptyOperator(task_id="task_3")
            task_1 >> [task_2, task_3]
        task_4 = EmptyOperator(task_id="task_4")
        task_5 = EmptyOperator(task_id="task_5")
        task_6 = EmptyOperator(task_id="task_6")
        task_7 = EmptyOperator(task_id="task_7")
        task_8 = EmptyOperator(task_id="task_8")
        start >> section_1 >> [task_4, task_5, task_6, task_7, task_8]
    dagrun = dag_maker.create_dagrun(
        run_id=run_id,
        execution_date=execution_date,
        state=State.FAILED,
        run_type=DagRunType.SCHEDULED,
    )
    def get_ti_from_db(task):